Hamlin, Hamlin & McGill is a successful law firm based in Albuquerque. It was started by Chuck McGill and Howard Hamlin's father. At some point Howard made senior partner as well. Also for a yet unexplained reason, Howard's father seemed to withdraw from the firm. It was then run by Howard and Chuck or just Howard (we are not yet sure if Howard's father withdrew before or after Chuck was forced to go on an indefinite sabbatical from the firm due to his illness). Chuck did go back to working in a limited capacity thanks to Jimmy invigorating him with the Sandpiper Crossing case.
Season 1 (Better Call Saul)
It is revealed in a flashback that Jimmy used to work here as a mail man. It is also revealed that Chuck was the one that didn't want him to work at HHM as a lawyer because he thought that Jimmy wasn't a real lawyer.