Secret Service agents perform two separate but equal functions: one part protective bodyguard, the other part criminal sleuth. Agents wield a lot of power when investigating current and potential criminal cases.
First, Secret Service agents don't just have a right to bear arms—they're actually supplied with them (source). Agents can execute warrants legally issued under United States law, and can arrest people without warrants if the alleged perpetrator commits an offense against the United States in front of the agent. An agent can even arrest someone if the agent reasonably believes the subject has committed that crime.
A Secret Service agent can post a reward for information or for other unspecified services, which can lead to the arrest of the person who allegedly violated the law. Regardless of what the agent has on their plate, they partner with the United States Attorney's Office to make sure their duties are performed in accordance with the law.
As for an agent's protective functions, those are probably not much of a secret to anyone—wear sunglasses, look intimidating, make sure nobody kills your boss.
However, you might wonder what power Secret Service agents can flex to keep the president—not to mention everyone else on the list—out of harm's way. It turns out they can do a whole lot of things.
Can the agent close major city streets around a presidential appearance? Yep. Can an agent set up a security perimeter, which might include snipers on rooftops and welded-shut manhole covers and mailboxes? They sure can. Secret Service agents can insist on maddeningly inconvenient security checkpoints, alterations in speaking venues, and pretty much anything else they feel is necessary.