Suppose I were to kill someone and was charged with manslaughter, found not guilty, and more evidence surfaces, does double jeopardy prevent me from even being charged again for that action (the killing), or does it only protect me from the charge of manslaughter again?
Under U.S. law, double jeopardy prevents you from being charged with the same charge twice, and also from being charged with any offense which is a lesser included offense of the charged offense, or a charge so substantially similar that for constitutional purposes it amounts to the same crime. Basically, the test is whether a prior acquittal would be inconsistent with a new criminal charge.
For example, even though there is an additional element of the crime of murdering a postal officer to the crime of murdering someone on federal property, double jeopardy would probably bar a retrial of a murder on federal property case simply because the victim happened to be a postal worker and that element wasn't charged in the original indictment. This is because the acquittal of the first murder charge would almost always imply a jury determination that a murder didn't take place which would be inconsistent with a murder of a postal worker charge.
On the other hand, a trial on a murder charge would probably not bar, for example, a trial on a burglary charge (which at common law involved trespassing with an intent to commit a crime), even if the burglary charge arose from the same conduct. This is because an acquittal on a murder charge isn't necessarily inconsistent with the existence of a trespass, or with the intent to commit some crime other than the murder for which the defendant was acquitted.
But the exact way that the line gets draw is tricky and while what I have described is a good general summary of the cases interpreting the double jeopardy clause, it isn't a perfect one. This issue has been litigated many, many tines over the years, so there are a lot of cases that are squarely on point addressing specific fact patterns in precedents that are binding case law that are not always a perfect fit to the general principles. In these circumstances, the binding case law is going to control, at least until a court with appellate authority over the court whose case established the precedent in question decided to overrule a prior precedent from the lower court, or in the case of U.S. Supreme Court precedents, until the U.S. Supreme Court revisits one of its own prior precedents as wrong decided or wrongly interpreted, which happens now and then, although it is a rare event.
Double Jeopardy, in the US, protects against a second trial for the same action. It is prohibited by the 5th Amendment to the US Constitution, which reads (in pertinent part):
... nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb;
As the Wikipedia article says:
As described by the U.S. Supreme Court in its unanimous decision concerning Ball v. United States 163 U.S. 662 (1896), one of its earliest cases dealing with double jeopardy, "the prohibition is not against being twice punished, but against being twice put in jeopardy; and the accused, whether convicted or acquitted, is equally put in jeopardy at the first trial".
However, when an action is a crime under both US Federal law and State law, a person may be tried separately by both the state and federal governments, and Double Jeopardy is not a bar to such a second trial.
Cases where added evidence comes to light after a trial are very much included in the double jeopardy bar, and the first trial (whether or not it resulted in a conviction or even a final verdict) prevents the second from being held.
As The Wikipedia article on the case says:
The guarantee against double jeopardy ... provided that where the defendant was acquitted of robbing one victim, the government could not prosecute the criminal defendant in a second trial for a different victim in the same robbery.