Just because Red Robin is one of my favorite mantles, and since you've already gotten good answers about Tim in the suit, here's some other people who have taken it up at various times (and in various timelines):
In Kingdom Come, Dick Grayson becomes Red Robin while working with Superman and the Justice League. This was the first appearance of Red Robin, and the suit can be seen as symbolic of how Dick has grown as both a person and a hero. The design may have also been symbolic of Dick's desire to mend his relationship with Bruce, while still maintaining his own identity. The theme of the young hero coming into his own would return more than once; from a certain point of view, becoming Red Robin is something of a rite of passage for the Robins.
During the Countdown to Final Crisis, the Jason Todd of Earth-51 takes on the RR mantle and wears it until the Crisis has ended, when he trades it for his Red Hood identity. Tim alludes to these events when he takes up the mantle to search for Bruce, who is presumed dead after the events of Final Crisis.
Before Tim got hold of the suit, however, it is found by Ulysses Armstrong, who wears it while committing various crimes and terroristic acts as a form of psychological warfare against Tim. He would later reveal himself to be the new, villainous Anarky. Tim also mentions this in the RR solo series, remembering a time when the RR identity was "tainted." Tim dons the suit during this time, but does not yet take on the identity fully.
That would not happen until Dick becomes Batman in the missing Bruce's stead. Dick chooses Damian Wayne to be his Robin, reasoning that Robin is meant to be a student of Batman. Dick sees Tim as an equal, not a student, and though Tim does not appreciate this decision at the time, he accepts it and uses the RR mantle to disguise himself while searching for Bruce. He alters the recovered suit somewhat, including the addition of a cape that allows him to glide, leading to the design you're probably familiar with.
As for why I like the Red Robins so much (if anyone is still reading this), for me, it perfectly represents the middle ground between the Robin identity and the mantles each Robin chooses for themselves as they mature. The Robins are some of my favorite comics characters, partly because we have seen them grow from Boy Wonders into true heroes in their own right. If anyone is going to take on the identity permanently, though, I'm glad it's Tim. Sure, Dick was first, but he became Nightwing for a reason and I'd hate to see that taken from him. Jason's relationship with the Robin mantle is complicated, to say the least, and he's worked just as hard at being Red Hood as Dick has worked at being Nightwing. But Tim... Tim wanted to be Robin more than anything. Wanted it so bad that he earned the right to do so without even being asked. But, Dick was right to refer to Robin as a "student position," and Tim is too good a hero to stay in that position forever. Change is an inevitable part of growing up, after all.