Although the term is frequently used in Japan to refer to a PC ("personal computer"), in English it refers primarily to the humanoid computers found in CLAMP's Chobits.
In CLAMP's Chobits manga, personal computers are effectively androids: they have an advanced A.I. and look like humans, completing all the same tasks as a personal organizer.
There are two primary examples of persocoms inherent in Chobits. The first is Chii, who is an atypical persocom in that her A.I. is so advanced that she can actually fall in love and learn new tasks without installing additional software.
The second is Sumomo (or Plum in the English manga), a mobile persocom that is small enough to fit in a pocket or in the palm of a hand-- the persocom equivalent of a laptop or PDA.