Bikur cholim means "visiting the sick." The root "bakar" means "to examine, distinguish" or "search," from which it comes to be used of "asking after someone's health," or "visiting" people who are unwell. It is also related to the word "boker," meaning "morning, light" or "salvation."
Bikur cholim is one of the basic ways in which we can perform chesed, "loving-kindness" for another person. The Talmud describes it as a way in which we can imitate God (Sotah 14a). A whole section in the Shulchan Aruch (Yoreh Deah, 335) guides us as to how best to fulfil this mitzvah. The more often one can visit, the better, as long as it doesn't start to bother the patient.
There is an argument in the halachah, about whether one ought to go to visit someone whom one dislikes when they are sick. Maharil (15th century) sees no problem with this. The Rama (16th century), however, says that one should not, in case it seems to the patient that you are gloating over his misfortune and he will feel worse.
According to the Shulchan Aruch, part of the mitzvah of bikur cholim is to pray for the recovery of the person one is visiting. In so doing we acknowledge that, even with the best doctors, and all the armfuls of flowers, chocolates and trashy magazines we can bring, healing is in God's hands.