| Nedarim
At first glance one would not expect a Talmudic discussion on the mitzva of bikkur cholim in the midst of a discussion of vows. However upon reflection of the nature of the human psyche such becomes readily understandable. "One who is forbidden by a vow to receive benefit from another and he enters to visit him [when the other is sick] he [the visitor] may stand but he may not sit" (Nedarim 38b).
As we see throughout Masechet Nedarim people when angry at others would often take a vow forbidding them from having anything to do with such a person. While such may do little to improve relations between people it does prevent the situation from deteriorating - a precursor to "good fences make good neighbours". And as often happens, our priorities shift when one is on their deathbed. We should not be surprised that people who not have spoken in years may want to try and make peace before it is too late[1]. How sad it is if even possible impending death is not enough to get people to "bury the hatchet" and make up. For this at least they should be applauded.
Nonetheless such a visit is limited "to standing" i.e. one of a few minutes, but he may not sit. The Gemara explains that we are dealing with a situation where the sick person may not derive benefit from the visitor and people were paid to sit at the bedside of the sick - not much different than hiring a night nurse to constantly be with the patient. By "sitting" at the bedside (in lieu of the nurse) the sick person is saving the money he would have otherwise had to pay to have another sit by the bedside (Nedarim 39a).
That visiting the sick is especially important when one does not get along with the person taken ill seems to be hinted at by the source of this mitzva. If asked, most people would presumably answer that the mitzva to visit the sick is derived from G-d Himself who visited Abraham after his brit milah. And in fact the Talmud uses such as an example of our duty to actualize our Divine image "just as He visits the sick, so too you should visit the sick"(Sotah 14a). However our Gemara quotes a much different source, apparently understanding that Gemara as referring not so much to visiting the sick per se, but to emulating G-d, which includes amongst other things visiting the sick. A specific reference to the mitzva of visiting the sick is derived from none other than Korach[2] (Nedarim 39b).
"If these men die the common death of all men, and be visited after the visitation of all men, then the Lord has not sent Me" (Bamidbar 16:29). Reish Lakish interprets the phrase "the common death of all" as referring to one who dies "the common death of all men, who lie sick in bed and men come in and visit them, what will people say? The Lord hath not sent me". It is only because Korach's death was both supernatural and sudden (and declared such in advance) not allowing for bikkur cholim that it was evident that it was a sign from G-d.
One visits his enemy hoping to turn him to a friend; and one visits friend due to the concern we have about them. The Gemara believed that those who have visitors often live longer. In the words of the Gemara "whoever visits one who is sick takes away 1/60th of his pain" (Nedarim 39b). When one knows that others care for them they have a greater will to live[3]. Rabbi Akiva goes so far as to assert "that whoever does not visit the sick, ke'eilo[4], it is like he is spilling blood" (Nedarim 40a).
Bikkur cholim involves more than visiting the sick. It requires we davenfor his well-being. When people are in need we express our faith that all that happens in this world - good and bad - emanates from the Creator[5].
[1] The Ramah (Yoreh Deah 335:2) disputes the ruling that an enemy may visit the sick or offer comfort to the family upon the death of his adversary.The Shach notes that each case must be evaluated individually.
[2] The Gemara uses the term remez meaning that this verse hints at the mitzva to visit the sick but the verse is not actually a direct command to visit the sick. The Rambam (Aveilut 14:1) does not codify a specific mitzva to visit the sick including it under the general rubric of gemilut chasadim, acts of kindness.
[3] In our recent trip to Poland we discussed how one of the factors that helped one's chances or survival was having others who took care of him or her. Moreover those who took care of others also had higher rates of survival. The knowledge that the lives of others depended on them helped them in their quest for survival.
[4] The insertion of ke'eilo indicates that the two are not to be actually equated. It expresses, perhaps with a bit of hyperbole, the importance of the matter under discussion. Other actions equated to spilling blood include "whitening one's face" (embarrassing another - Bava Metzia 58b) and willfully not having children (Yevamot 63b).