Career is the term generally given to the tributes from District 1, District 2 and District 4 during the Hunger Games. They team up together to kill off the other tributes, and generally one of them wins. The people of these districts also tend to train their children for the Games from an early age, even though training officially is against the rules. They never get in trouble for it, though.
Not only are these tributes often better with weapons since they have illegally trained, they come from the wealthier districts and are often better fed and therefore stronger than the tributes of other districts.
While Districts 1, 2 and 4 make up the usual Career pack, occasionally a tribute from another district who is particularly strong or skilled in some way is able to join them. This is seen in Catching Fire when Brutus - who is from District 2 - requests Katniss Everdeen as an ally after seeing her superior shooting abilities. While she turns down his offer, it was indeed a chance to be a Career. Peeta Mellark is also able to become part of the Career pack during the 74th Hunger Games in The Hunger Games. However, according to Haymitch Abernathy Peeta barely got in with the Careers, since he waited until the Games started. Apparently the Career pack is normally determined before they begin, likely during training.
 The 74th Hunger Games
During the 74th Hunger Games the Career pack consisted of Glimmer and Marvel from District 1, Clove and Cato from District 2, the District 4 female tribute and the District 3 male tribute. The District 4 tribute likely would have been part of the Career pack, but he was killed during the initial battle at the cornucopia. The District 3 boy was only allowed into the alliance because he was able to reactivate the mines to help protect the Career camp - keeping them, their food, and their supplies safe. Peeta Mellark was also originally a part of the pack, pretending to want to kill Katniss Everdeen. However, when he saves her from Cato after she sets tracker jackers and them all, his cover his blown and it nearly kills him.
Out of the Career pack, both Glimmer and Marvel are killed by Katniss. Glimmer during the tracker jacker attack, and Marvel from one of Katniss' arrows after he spears Rue. Clove is killed by Thresh during the Feast, and Cato faces a long and drawn-out death initiated by the mutts in the final hours of the Games, but is finally shot by Katniss out of mercy towards him. The female tribute from Four also died in the tracker jacker attack.
 The 3rd Quarter Quell
During the 3rd Quarter Quell, the Career pack is made up of Cashmere and Gloss from District 1 and Enobaria and Brutus from District 2. Mags and Finnick Odair were the tributes from District 4, but they didn't join the Career pack like the tributes from their district normally do. Mags was likely too old to be considered useful, and Finnick deliberately chose Katniss Everdeen instead. Finnick and Mags both knew about the plan to get the tributes out of the arena, while the other Careers did not and carried on as usual.
Before the Quell, Peeta Mellark, Katniss and Haymitch Abernathy all trained like Careers. And even though it was illegal, as with the Careers in other years nobody ever stopped them. It made them a lot stronger, and ready to face their competitors. After all, they knew Katniss and either Peeta or Haymitch were re-entering the arena.
During training, Katniss seems disgusted that Haymitch wants them to join the Careers, and Haymtich points out that their plan had been to train like Careers. In fact, Brutus does request Katniss as an ally after seeing her shoot, but she turns it down.
During the Quell the Career pack attacks Katniss, Peeta, Finnick, Johanna Mason, Wiress and Beetee when they are on the small beach the cornucopia was located on. Seconds after Gloss slits Wiress' throats Katniss is alerted of their presence (because Wiress stopped singing) and shoots the District 1 tribute in the temple. While Katniss is distracted with Gloss, Johanna buries one of her axes deep into the chest of his sister Cashmere. While Brutus and Enobaria escape that attack, Brutus does die minutes before the escape attempt (possibly crushed by large debris of the dome after Katniss destroyed it with an electrified arrow). Enobaria is captured and taken to the Capitol along with Johanna and Peeta, although she is not likely tortured as the other two (and Annie Cresta) are.
The Careers are also mentioned in relation to the 2nd Quarter Quell. These were the Games Haymitch Abernathy was both a tribute in and victor of. At one point Haymitch found himself in a fight against three of the Careers. He had killed two and the third was about to slit his throat when Maysilee Donner killed him with one of her poison darts. This was the birth of the alliance between the two that lasted quite awhile. However Maysilee eventually died, and finally it was just Haymitch against one of the female Careers. It was a very bloody battle that involved heavy injuries on both sides. The Career threw her axe at Haymitch, but it flew out of the arena, leaving them both weaponless (Haymitch had lost his just prior to this incident). It was then pretty much up to whoever of the two could survive their injuries longer than the other, at least in the mind of the Career, and she thought she might have a chance with her missing eye against Haymitch's gut-spilling injury. However, Haymitch knew the trick about the arena boundaries, and was expecting the axe to be rebounded back. The Career however had no idea, and the axe flew back into her, effectively killing her.