The Shadow Broker is in the business of buying and selling information. He is a shady figure in the politics around The Citadel, only working through agents. His species were the Yahgs, and some even believe him to be multiple people working as a singular figure. He is a master of his trade, and the information he buys and sells never gives an upper hand to the buyer or seller, guaranteeing his business' stability. He is an influential and powerful figure in the Mass Effect universe, and in Mass Effect 2 , Liara T'soni, your former teammate, is on her own mission to find and kill the Shadow Broker. He then appears in the Mass Effect 2 DLC "Lair of the Shadow Broker" where Liara T'Soni and Commander Shepard team up to defeat the Shadow Broker.
Mass Effect 1
One of the agents of the Shadow Broker has an office in the Financial District in the Citadel. When talked to he explains some of his theories on who the Shadow Broker is, and what exactly he does.
Tali'Zorah nar Rayya came to the Citadel in search of the Shadow Broker. She had a data file fished off of a Geth she had killed that proved Saren Arterius to be guilty of treason. Tali is double crossed by the agent she was asked to work through, Fist, and the information never reaches the Shadow Broker.
In the mission Hades' Dogs, Shepard finds data files after raiding the Cerberus' base. An agent of the Shadow Broker then contacts Shepard when he is back aboard the SSV Normandy, offering to buy the data files from Shepard.
Mass Effect: Redemption Comic
The Shadow Broker and Liara T'Soni are both looking for Commander Shepard's body after he has died at the hands of the collectors. T'Soni is working for Cerberus and has the help of a Drell named Feron. While T'Soni captures Shepard's body, Feron is captured by the Shadow Broker and attracts the ire of T'Soni. This leads into the DLC in Mass Effect 2 called "Lair of the Shadow Broker".
Mass Effect 2
In the "Lair of the Shadow Broker" DLC for Mass Effect 2, The Shadow Broker turns out to be of an alien race called the Yahg. The Yahg had killed the previous Shadow Broker and had thus become the new Shadow Broker. He attracts the ire of Liara T'soni after he captures her friend Feron and she teams up with the newly re-animated Commander Shepard to take down the Shadow Broker.
After discovering his lair and defeating him in battle, T'Soni and Shepard both hear the cry for contact from his information network.
Rather than destroying his ship, and its database and network within, Liara T'soni assumes the role of the Shadow Broker to aid Shepard in his continued quest against the reapers.