Brief detentions and reasonable suspicion
You can be briefly detained by police if they have reasonable suspicion that you committed a crime. Terry v. Ohio, 392 U.S. 1 (1968)
What reasonable suspicion "means" can only be fully understood by reference to subsequent case law (which I will expand this answer to do), but as a basis, the court said in Terry that:
the police officer must be able to point to specific and articulable facts which, taken together with rational inferences from those facts, reasonably warrant that intrusion
This standard has been reiterated as recently as in Heien v. North Carolina 574 U. S. ____ (2014), where they say "All parties agree that to justify this type of seizure [a traffic stop, in the case of Heien], officers need only reasonable suspicion — that is, a particularized and objective basis for suspecting the particular person stopped of breaking the law" (internal quotation marks omitted).
The reasonable suspicion standard was also used recently in Navarette v. California 572 U. S. ____ (2014). They reiterated that reasonable suspicion is dependent upon both the content of information possessed by police and its degree of reliability, quoting Alabama v. White, 496 U. S. 325, 330 (1990). A mere "hunch" does not create reasonable suspicion, but the level of suspicion required by the reasonable suspicion standard is "obviously less than is necessary for probable cause".
Arrests and probable cause
To be arrested, police require probable cause. Brinegar v. United States, 338 U.S. 160 (1949)
In more detail, probable cause exists (from Brinegar v. U.S.):
where the facts and circumstances within the officers' knowledge, and of which they have reasonably trustworthy information, are sufficient in themselves to warrant a belief by a man of reasonable caution that a crime is being committed
The rule of probable cause is a practical, nontechnical conception affording the best compromise that has been found for accommodating these often opposing interests. Requiring more would unduly hamper law enforcement. To allow less would be to leave law-abiding citizens at the mercy of the officers' whim or caprice.
As in the case of reasonable suspicion, the probable cause analysis is case-by-case and fact-intensive, so to understand the contours of probable cause will require reference to much subsequent case law.
In Beck v. Ohio, 379 U.S. 89 (1964) the question before the court was entirely "whether or not the record in the case before us can support a finding of probable cause for the petitioner's arrest". In that case, it turned out that the information they had received about the arrestee was not sufficient for probable cause, but regardless, the test the court applied was whether the police had probable cause for the arrest.
While I am confident in the correctness of this answer, what each of these standards means will take hours of work to flesh out, which I plan to do. The courts have repeatedly reiterated and referred to these decisions/standards, but the analysis is very fact-intensive and is done case-by-case. Also, I realize the presentation is a little scattershot, as I'm first just looking to include relevant cases and statements the court has made about these standards, but I'll re-make it into a coherent story every once in a while.