What are the “horrible heads” in Mr. Jaggers’ office? Why do you suppose he keeps them?

The "horrible heads" are casts, material that has been heated in a mold. In this instance, the heads are "busts" or "death masks" of Mr. Jagger's most famous clients.

"These?" said Wemmick, getting upon a chair, and blowing the dust off the horrible heads before bringing them down. "These are two celebrated ones. Famous clients of ours that got us a world of credit. This chap (why you must have come down in the night and been peeping into the inkstand, to get this blot upon your eyebrow, you old rascal!) murdered his master, and, considering that he wasn't brought up to evidence, didn't plan it badly."

We can only infer he keeps them as remembrances.... the men he represented and couldn't save.