Brynden Tully is Catelyn Stark's uncle and the younger brother of Hoster Tully, the recently deceased Lord of Riverrun. If you've seen the most recent GOT episode, you know that Brynden possesses strong archery skills. Those, in addition to other fighting skills, make him a respected knight and commander.
In the books, we meet Brynden when Catelyn is carrying Tyrion to the Vale and her crazy sister Lysa Arryn. He's serving as Knight of the Gate (he's basically in charge of making sure no unwelcome visitors get up the mountain to Lysa. It's supposedly a great honor). When he hears about the Lannisters attacking the riverlands (his family's lands) he resigns his position and leaves with Catelyn to meet up with Robb Stark and his army. Once there Robb immediately makes Tully part of his war council and puts him in charge of the outriders. While in that capacity he does a lot of good work that greatly helps Robb's army.
The reason he's known as the Blackfish is related to the reason he was in the Vale, instead of at Riverrun all along. According to the books, Brynden and his brother Hoster had a major falling out when Brynden refused to accept a marriage Hoster had arranged for him. At one point Hoster called Brynden the black goat of the family, but instead of taking it personally Brynden pointed out that since the Tully's sigil is a fish he'd actually be the black fish of the family. As such, he took the name Blackfish and changed his personal sigil to a black trout. He also went to the Vale to serve Lysa and her husband Jon Arryn (the guy whose death basically started the whole story).
The Blackfish is definitely a cool character who doesn't take crap from anybody. He's good at what he does (leading armies and fighting), and he knows it. Because of his brother's death* he should be sticking around Riverrun to make sure things stay in order, which means he should remain part of the main story.
*By the way, in the books he reconciled with his brother before Hoster passed away.