What is equity and/or an equitable remedy? How does it relate to the common law?
In legal systems descending from English law, the term equity refers to an aspect of those legal systems that has origins distinct from positive law and the traditional common-law writs. Equity traces its origin to the Court of Chancery. Much of what follows can be found in Snell's Equity. Equitable remedies (equitable relief) are those remedies available through rules and principles of equity, today applied by courts of unified jurisdiction.
Equity as part of law: equity is certainly part of law in the sense that it is part of the legal system, applied by courts, and can result in equitable remedies. "Equity in this sense has been a feature of many legal systems from ancient times" (Snell's Equity, §1-002).
Equity versus law: equity is distinct from "law" (or common law) in the sense that equitable remedies are meant to supplement "legal rights."
The common law and equity operated in their own areas of specialisation. Even in cases where common law and equitable rules touched on the same subject matter, the rationale of equitable intervention was not to conflict with the common law. The Chancellor's jurisdiction over the parties was in personam and enforceable by specific remedies, unlike those of the common law which were generally for the orders for the payment of money. (Snell's Equity, §1-012).
Legal and equitable remedies: "The division between legal and equitable remedies is an accident of history which cannot be explained in any other terms. No defining characteristic sets the remedies developed in Chancery, as a class, apart from the remedies developed by the common law courts." (Snell's Equity, § 14-001)
History and the present
A well-accepted unifying feature of "equity" and "equitable remedies" is their origin in the Court of Chancery of specialist equitable jurisdiction (Snell's Equity, §1-003). Even though today, this distinction in jurisdiction has been abolished by the Judicature Acts of 1873 and 1875, equity still means that body of law with its origins in Chancery. While equitable rules and principles have certainly continued to evolve, they all trace their pedigree to the jurisdiction of the Court of Chancery (Snell's Equity, § 1-004).
Equitable doctrines have retained their distinctiveness as "secondary" to legal rights. They impose an overlay: that "primary legal rights should be exercised consistenly with good conscience" (Snell's Equity, § 1-033).
This view from Canson Enterprises Ltd. v. Boughton & Co.,  3 SCR 534 highlights the messiness and debate about the confluence and remaining distinctiveness of equity and law:
I am aware that reservations have been expressed in some quarters about this fusion or, perhaps more accurately, mingling of law and equity... But no case was brought to our attention where it has led to confusion, and there are many cases, some of which I have discussed, where it has made possible a just and reasonable result. It simply provides a general, but flexible, approach that allows for direct application of the experience and best features of both law and equity, whether the mode of redress (the cause of action or remedy) originates in one system or the other. There might be room for concern if one were indiscriminately attempting to meld the whole of the two systems. Equitable concepts like trusts, equitable estates and consequent equitable remedies must continue to exist apart, if not in isolation, from common law rules. But when one moves to fiduciary relationships and the law regarding misstatements, we have a situation where now the courts of common law, now the courts of equity moved forward to provide remedies where a person failed to meet the trust or confidence reposed in that person. There was throughout considerable overlap. In time the common law outstripped equity and the remedy of compensation became somewhat atrophied. Under these circumstances, why should it not borrow from the experience of the common law? Whether the courts refine the equitable tools such as the remedy of compensation, or follow the common law on its own terms, seems not particularly important where the same policy objective is sought.
Even though equity has developed as supplemental and secondary to the common law, "in a conflict between the two, equity would prevail" (Snell's Equity, §1-010). For example, where a contract would be enforceable by its terms at common law, an equitable remedy developed called recission that allows for the "extinction of a contract and the resotration of the parties to their original positions" (Snell's Equity, § 15-001).
This has been codified in some jurisdictions. See British Columbia's Law and Equity Act, s. 44:
If rules of equity and law conflict, equity prevails
44 Generally in all matters not particularly mentioned in this Act in which there is any conflict or variance between the rules of equity and the rules of the common law with reference to the same matter, the rules of equity prevail.
Where this distinction can matter
This distinction can still matter. As just one example: "Different rules of following and tracing have evolved at common law and in equity" (Snell's Equity, § 30-052). See Citadel General Assurance Co. v. Lloyds Bank Canada,  3 SCR 805 at para. 58:
In my view, a distinction should be made between the imposition of liability in “knowing receipt” cases and the availability of tracing orders at common law and in equity. Liability at common law is strict, flowing from the fact of receipt. Liability in “knowing receipt” cases is not strict; it depends not only on the fact of enrichment (i.e. receipt of trust property) but also on the unjust nature of that enrichment (i.e. the stranger’s knowledge of the breach of trust). A tracing order at common law, unlike a restitutionary remedy, is only available in respect of funds which have not lost their identity by becoming part of a mixed fund. Further, the imposition of liability as a constructive trustee is wider than a tracing order in equity. The former is not limited to the defence of purchaser without notice and “does not depend upon the recipient still having the property or its traceable proceeds”; see In re Montagu’s Settlement Trusts, supra, at p. 276.
If a remedy is understood to be a legal right, then the innocent/wronged party is owed that right once the wrong is recognized by a court. However, if the remedy is understood to be equitable, no one ever has a claim as of right to the remedy, and it will always be subject to the discretion of the court to grant the remedy even if accepting the wrong underlying the claim.
Some equitable remedies
Wikipedia gives one list of equitable relief consistent with much of what is in Snell's Equity. Here I'll only list some of those that you'll most likely recognize:
- injunctions (Snell's Equity, § 18)
- specific performance (Snell's Equity, § 17)
- rescission (Snell's Equity, § 15)
- rectification (Snell's Equity, § 16)
- tracing (Snell's Equity, § 30-051)
- equitable estoppel (as a family of related doctrines), as distinct from common law estoppel (Snell's Equity, § 12)
Maxims of equity
Wikipedia also lists various maxims of equity also consistent with much of what is in Snell's Equity, § 5). I'll only highlight a few that are still widely relevant and known:
- "He who comes into equity must come with clean hands"
- "Equity aids the vigilant and not the idolent" (the principle behind laches)
- "Equity looks on as done that which ought to be done" (the principle behind equitable assignment)
@Jen does a good job of recapping the historical antecedents of equity and some of its most notable features. I'll add a few general observations and some additional observations particular to the United States experience.
Like the common law, equity jurisprudence arose largely through case law rather than through detailed statutory guidance, unlike the legal tradition of civil law countries. Civil law countries include the legal systems of most of continental Europe and most of Latin America, Quebec, Japan and South Korea, to name a few.
In cases at law, if a claim is established in a lawsuit, the award of damages is generally non-discretionary for the judge or jury hearing the case.
In cases at law where the main remedy is a money judgment, enforcement of that judgment through judgment liens on real property, writs of execution directed at personal property, and writs of garnishment directed at private rights of a judgment debtor in the control of a third-party, are the norm.
The right of people to sue the government for money damages on claims arising in law has historically been tightly regulated and limited.
In claims arising in equity, even if one establishes the elements of a claim for relief to a judge, that is not generally the end of the story. The judge usually has wide discretion over precisely which remedy to provide in a case arising in equity and over whether to impose it in a case that nominally seems to quality for a grant of relief. The judge's call on these issues can only be reviewed for an abuse of discretion.
Equity claims rely much more heavily on the power of the court to hold parties in contempt of court (and hence subject to incarceration and/or fines) for failure to obey court orders.
The right to people to obtain equitable remedies such as injunctive relief against the government has historically been much broader and less restricted than their ability to obtain money damage awards from the government.
Jury Trial Rights
In the United States, the constitutional right to a jury trial in federal court hinges upon whether the claim for relief is one that would have been asserted in the courts of law in 1791, when the 7th Amendment to the United States Constitution was adopted, or in the courts of equity. If the claim is an equitable one, there is no right to a jury trial. In the case of newly created claims the question is whether the newly created claim (usually statutory) is closer to a claim for money damages under common law, or a claim arising in equity in 1791.
Many states also make this distinction even though the 7th Amendment right to a civil jury trial does not apply to U.S. states and is not even a state constitutional right in some states.
Some common claims in which there is no right to a jury trial include: foreclosures, divorces, probate matters other than will contests, disputes within partnerships, disputes within corporations and similar entities, claims seeking any of the equitable remedies identified by @Jen other than claims to obtain physical possession of property, partitions of real property, quiet title actions, mechanic's lien enforcement, matters related to trusts, constructive trust claims, fraudulent transfer claims, child custody, and adoption.
In contrast, claims for breach of contract, and claims for money damages in tort will generally give rise to a right to trial by jury.
There is a split of authority between U.S. states over whether equitable defenses like laches and unclean hands apply only to claims arising in equity, or can also be used to defeat claims arising in law. The majority rule is that equitable defenses are available only to equitable claims. But, some states, such as Colorado as recently clarified in Hickenson v. Vessels (Colo. 2014), allow equitable defenses to legal claims.
Incomplete Mergers Of Law And Equity
Most U.S. courts (including the Article III federal courts in the U.S.) have fully merged courts of law and courts of equity administratively, but the merger is not entirely complete in some U.S. states.
The state of Delaware and Wyoming, for example, have Courts of Chancery that specialize mostly in cases that were historically in Chancery Court jurisdiction (especially cases involving the internal affairs of business entities) although the division is not absolute since other courts in these jurisdictions can also apply equitable principles and since the modern courts of chancery don't have jurisdiction over cases exactly identical to the historical English Court of Chancery.
Other states have specialized courts that have jurisdiction over cases that were historically mostly the province of equity courts, like the Surrogate's Courts in New York State which have jurisdiction over many equitable matters involving trusts and decedent's estates.
Many courts of limited jurisdiction in the United States lack subject-matter jurisdiction to grant equitable remedies such as injunctions and specific performance, and many U.S. courts of limited jurisdiction also lack jurisdiction to rule in cases where title to real property is at issue, even if they have jurisdiction over evictions.
Historically Non-Common Law Jurisdictions In The United States
The State of Louisiana and the Commonwealth of Puerto Rico did not have common law legal systems when they joined the United States, and the civil law systems of the French and Spanish respectively that were used in these jurisdictions never had a law-equity distinction in the first place.
Much of the Southwest United States was part of the Republic of Mexico before it became part of the United States, and civil law concepts, particular in the areas of marital property rights and rights in real property carried over somewhat into modern law in these places (despite these states largely adopting most other common law legal concepts). Again, these concepts arose in a legal system than did not the law-equity distinction so the way that concepts like law and equity are applied to these concepts is haphazard.