Axel is a very popular character from the Organization. For most people, he's even their favorite member, and why not? He's funny, cool and has a pretty dominant role in three Kingdom Hearts games.
Lea in Birth by Sleep:

As we know, a Nobody is created when someone who has a very strong will succumbs to the darkness and becomes a Heartless. The body that is left behind becomes reanimated, but it doesn't have a heart, thus a Nobody is born.
Lea is the Somebody to Axel's Nobody. The only time we see Lea is during Ventus's story. Ven is feeling very down and thinking that he doesn't have any friends and is sitting around Radiant Garden when Lea and Isa (Isa being Saix's Somebody) come walking by and see him holding his makeshift Keyblade.
After a bit of talking Ven and Lea have a battle of sorts, which we don't actually get to witness, and Ventus appears to be the winner. The two talk a little bit longer and say that they are friends now.
Axel in (Re:)Chain of Memories:
Appearing in quite a few different key moments within the game, Axel has a major role within these games. The first "boss battle" in the game for Sora is fighting Axel.
As seen in 358/2 Days Axel and the rest of the Organization that goes there, are sent to capture the keyblade wielder. But during their time there, Marluxia is trying to gather some dissenters in the Organization and take over the rest of the Organization. Axel, being a double agent, helps put a stop to Marluxia and in the end Sora is able to deafeat the pink-haired member because of it.
A cool side note, Axel is the only Organization member to appear in both Sora's story and Riku's.
Axel in 358/2 Days:
This is probably Axel's biggest role within the series. In KH2 we are able to glean that Axel and Roxas are best friends, but how come and why, we weren't actually able to discern. So with the installment of 358/2 Days we saw everything that transpired between them. Literally every day (except those at the very beginning and when Axel was at C.O. and lost) Axel went to the top of Twilight Clock Tower and had ice cream with Roxas, and eventually Xion as well.
There is a lot of character development that goes on in these short conversations between Axel and his friends. But one of the biggest revelations from the game is that Axel appears to have feelings, but not only that a heart. No official reason behind this exists, but it is to be believed that a major reason Axel acts the way he does is because he's so close to Roxas, who in fact does have a heart, so he's "borrowing" the feelings from Roxas, when he can't do it himself.
Axel in Kingdom Hearts II:
In the beginning of the game when you're still playing as Roxas, Axel is very predominant. When this game came out, nobody knew who Roxas was, or what Axel had to do with him, so it was really confusing. Having this dude chase you and tell you things like, "I'm your best friend," was strange. But you couldn't help but like the guy.

Within the first couple of hours as Roxas you are actually forced to fight Axel two different times, and the second time is pretty amazing. It reveals the power Axel really has and that, yeah, he really is cool.
After the intro with Roxas, Axel kind of disappears for a while. There are a couple of cut scenes with Pluto running around through Destiny Islands and The World that Never Was, finding Axel and taking him to Kairi, but beyond that we don't really see much.
The end of the KH2 is where Axel is probably the coolest character you've ever seen. Sora is trapped in a room with thousands of Dusk Nobodies and you have Axel fighting with you to defeat all of them. Eventually Axel goes all out and uses an amazingly powerful attack and destroys the rest of the Dusks. This pushes Axel to the limits and causes him to exhaust himself enough that he actually dies.