Liam Gallagher is a recurring character. He is the youngest Gallagher sibling. Although the opening of the pilot implies that it's very likely that Liam is fathered by Frank's AA Sponsor, Season 1's episode Nana Gallagher Had an Affair reveals (through a paternity test) that Frank is truly Liam's biological father, despite both parents being white with a distant African-American relative.
Liam has served little purpose throughout the series due to his young age, however, he has been an important plot point in many instances. Monica tried to steal him to live with her and her girlfriend Roberta, however, the Gallagher siblings rallied to keep Liam with them. In Season 2, Liam was taken as collateral when Frank bet 10,000 dollars he could not pay back. In Season 3, Fiona gained legal guardianship over the Gallagher siblings, including Liam.
In Season 4, he got into cocaine left out at the Gallagher home, overdosed, and nearly died. However, he made it to the hospital in time. Subsequently, Fiona was arrested, jailed, and tried for child abuse and possession of an illegal substance. Due to his exposure to a controlled substance during his adolescence, he could possibly have brain damage, leading to developmental disorders.
In Season 8, Liam is attending the private school Frank was able to get him into in the previous season, but he's not getting the private school education. Liam is routinely taken out of class to be used as a token black kid while the private school gives tours to perspective parents, making the private school seem "diverse."