Just to add to
Fay's awesome post...
Hotch : We met his brother during the series, and were told that their father died quite young. Aaron therefore took that role within his family. It's a role he also takes on within the BAU, but when Elle calls him "Dad" one episode, he doesn't like it. Hotch is usually quite serious, and doesn't often let his guard down. He's also quite ambitious, although you get the impression it's because he wants to change the FBI for the better.
Gideon : At the beginning of the series, Gideon is on medical leave. Six agents were killed by the serial bomber, Adrien Bale, because Gideon trusted Bale when he shouldn't have. This one event affects Giden for the whole series, because of the guilt he still has over it. It's also the cause of tension between him and Morgan. (see next bit)
Morgan : Morgan was meant to be in the team who went out to catch Bale. He ended up not going, and surviving as a result. He still questions Gideon's ability to work in the BAU, and this sometimes leads to him questioning Gideon's leadership. Hotch frequently has to step in and have words with Morgan.
Morgan also argues with Reid, in a much more brotherly way. He teases Reid something rotten about his social awkwardness and uber-intelligence, but he's also the one who offers Reid advice and guidance for being "normal". Elle sometimes gets involved in this also - she's an old friend of Morgan's and the two regularly tease each other. His most amusing relationship is with Garcia though - she's the only one in the team who can consistantly out-snark him.
Work-wise, Morgan's the one who you'll see trying to "think" like the killer. Putting himself in their place and asking "so I'm a killer, what would I do here?"
Elle : Elle wasn't in the team at the beginning. We see her join a few episodes in - she wants to be part of the BAU and goes all out to impress. As previously said, she's an expert in sexual crimes and she really shines in cases where rape is involved. She's sensitive and thoughtful, but also extremely good at her job - we see her earn a lot of respect as the series goes on.
Reid : Is everyone's favourite nerd :lol: He's a walking encyclopedia, and he has a tendency to spout out facts at any given time, even when it's inappropriate. He's not doing it deliberately, Reid just can't read social situations and Hotch or Gideon have to gently guide him or explain him to members of the public. A lot of Reid's development is about the team teaching him to socially interact. He also breaks cases a lot of the time, thanks to Gideon teaching him to think outside the box. Gideon and Reid's father/son type relationship is probably the most important in the show. He's awkward but sweet and the team seem to try to protect him.
JJ : Has hidden balls of steel. Honestly, JJ is the one who takes most of the crap thrown at the team by law enforcement and the unimpressed media/public. She's the one who gives out or withholds information, and organises press conferences. She's exceptionally pretty, but also calm and level-headed. (I'd totally lose my temper half of the time, if I was her).
Garcia : Is probably the most under-appreciated character in TV history. She's their IT/tech expert, and she does everything from tracing phonecalls to hacking goverment websites. What she's really known for though, is her wise-cracking mouth. Almost everything that she says can be taken the wrong way, sounds smutty and is totally hilarious. She has no respect for rank, being as rude to Gideon and Hotch as she is to Morgan and Reid. She's especially flirty/snarky with Morgan, because he gives her equal amounts of sass back.
Garcia doesn't move from her office, which is filled with monitors and keyboards and the like. She is incredibly protective of her space - she hates her room getting messy or people bringing food in there. She's also a tetris addict
