Generally, when they see a crime occur, are "told" to by the d.a. or a warrant to, or have enough probable cause to cuff and stuff.

Yours is an awfully "open" question ... can you provide any more details on what you are looking for?

if they have enough evidence

Let's say your in the suspects situation.
When the police see you commit a crime and have physical evidence like seeing you spraypaint a wall or break into a store or even if you have anything that may link you to the scene of the crime such as a random object like a crowbar they can make an arrest.
They can also hold you for up to 24 hours at which pioint you will most likely interrogate you.
If you are interrogated SHUT UP. Don't say anything...don't even give them a story. They are also allowed to lie during interrogations.

This is an extremely complex area of the law. Some law students spend a whole semester on the law of arrests and searches - but I can break it down generally:

Generally, a police officer cannot make an arrest without a warrant.

A police needs to obtain a warrant from a judicial officer certifying the existence of probable cause to believe the individual to be arrested has committed a crime.

The exception to the rule is that police officer may arrest you if he has probable cause to believe a crime has been committed and exigent circumstances exist. Exigent circumstances is a very broad term that can encompass many factual scenarios. Generally though, they are circumstances which justify a police officer from deviating from compliance with the warrant requirement. For example, the officer may witness the crime first hand and it would be simply impractical or unworkable to get a warrant before making an arrest. This could be something simple, even a minor traffic infraction in some states. He may have received information - usually from a witness flagging him down or calling 911, and the risk of danger to civilians is great enough to outweigh the interest in compliance with the warrant requirement. Contrast this with a murder investigation: the crime has already occurred, perhaps occurring several days before. No one's life is directly in danger. There aren't any exigent circumstances that would justify foregoing with the warrant requirement.