The Stark bannermen are those noble lords of the north that follow the Starks; in this case, the guys that follow Robb Stark in to battle. They include all of the houses that are listed in the Appendix in the Stark entry: Karstark, Umber, Flint, Mormont, Hornwood, Cerwyn, Reed, Manderly, Glover, Tallhart, and Bolton. (Sounds like the most dangerous law firm ever.)
It may be hard to keep those straight. Well, just imagine how much harder it would be if you had to memorize their family symbols, like Bran (54 Bran 6.7). In any case, we've provided a little cheat sheet here of who fights from these noble houses. Also, to remind us of the cost of war, this guide tells you where these characters die or are captured. (The two major battles with the Stark army are at the Green Fork, which we see from Tyrion's POV, and in the Whispering Wood, which we see from Catelyn's POV [63 Tyrion 8, 64 Catelyn 10].)
Rickard Karstark and his three sons Eddard, Harrion, and Torrhen fight for Robb; Eddard and Torrhen are killed by Jaime in the Whispering Wood (64 Catelyn 10.60); and Harrion is captured in the Battle of the Green Fork (63 Tyrion 8.138). So Rickard gives up a lot for this fight. Hope it's worth it.
The Greatjon Umber and his son the Smalljon. The Greatjon (or just Jon to his friends) is a giant who understands force and has an odd sense of humor about it. For instance, after Grey Wind bites off two of his fingers for aiming a sword somewhere near Robb and Robb makes kind of a joke, Greatjon (minus two fingers) laughs about it. He's the first person to proclaim Robb the King in the north. So Greatjon, besides being huge and helping to fight, also helps us see how Robb has grown up.
Maege Mormont and her daughter Dacey are the kind of warrior-women that Arya probably wants to be. (You may recognize the name Mormont as well: Maege is Jeor Mormont's sister and Jorah Mormont's aunt. The only reason Maege is the head of the household is that her brother is in the Night's Watch and her nephew is in exile.)
Halys Hornwood (who dies at the Green Fork) and his son Daryn Hornwood (who dies in the Whispering Wood).
Medger Cerwyn (captured at the Green Fork).
Howland Reed remains at a castle to guard against any armies slipping up to the north; other than Eddard, he was the only survivor of the mission to save Lyanna Stark.
Wyman Manderly remains at home to guard the north (and also because he's tremendously fat); his son Wendel guards Robb at the Whispering Wood, while his son Wylis is captured at the Green Fork.
Galbart Glover and his younger brother Robett Glover.
Helman Tallhart is left with Walder Frey to make sure that the old man doesn't stab anyone in the back.
Roose Bolton is a frightening man whose castle, the Dreadfort, is supposed to have a room where they display the skins of their enemies. Yikes.
Although Robb Stark should only inherit Winterfell and command the northern lords, several Riverland lords join in at the end and swear fealty to Robb as their king. (Catelyn notices that these are noble "houses who had never been ruled from Winterfell" [72 Catelyn 11.103], but who are now swearing allegiance all the same.) Why don't they just call Robb "the King in the north and of the Riverlands"? We'll never understand these people.
In any case, among those who fight with Robb and swear fealty to him in 72 Catelyn 11 are Lyman Darry, Stevron Frey, Jason Mallister, Jonos Bracken, Tytos Blackwood, Marq Piper, and Karyl Vance.