Callandor appears to be simply a sword made of crystal, but is in fact a powerful sa'angreal that can increase the strength of a male channeler while they hold it (which is why it is sometimes referred to as "the Sword That Is Not a Sword"). Callandor was located in the Stone of Tear for thousands of years after the Breaking of the World, protected so that only the Dragon Reborn could remove it, an important sign of the Dragon's rebirth according to the Prophecies of the Dragon. Rand al'Thor uses it multiple times to perform great feats with the One Power in the novels once he draws it (therefore confirming even to himself that he is the fulfillment of prophecy, something he was unsure of.) However, he also leaves it in Tear when he leaves as a reminder to the Tairen people that he will be back.
The sword has since been removed from the Stone on Rand's orders by Narishma. The Prophecies state that he who draws it out must follow after, presumably indicating that Narishma will play an important role after Tarmon Gai'don. After the Cleansing of Saidin, it is presumed that Narishma kept the blade, but this is not confirmed. Its whereabouts are uncertain, which could lead to disaster.
Rand is told by Cadsuane Sedai that Callandor requires two women and one man to be able to use it properly, with one woman controlling the sword itself. If the sword is used by one man alone, it lacks the safety buffers that prevent him from burning himself out or killing himself, and the sword actually increases the Dark One's taint, causing temporary insanity. (This is a point of contention amongst readers, as to whether the sword increases the Taint or the man's susceptibility to the Taint; however, in the end, either interpretation has the same result.) Fortunately this risk may have been somewhat lessened by the Cleansing of Saidin.
Alternately, since all other angreal have this buffer, and can be used together, a man could conceivably use a small angreal (e.g. Rand's long-lost fat man statuette) in addition to Callandor, and be at no risk.
There is some confusion as to whether Callandor is a male-only sa'angreal or one that can be used by both sexes. Siuan Sanche tells Nynaeve that with Callandor in her hands, she could level an entire city with one blow, which was probably a metaphor. In addition, Lanfear expressed similar sentiments about the "Sword That is Not a Sword" in the Stone of Tear, but also states it would be useless for her.
The weapon is similar to the two swords of legendary King Arthur: Excalibur and "the sword in the stone", of Arthurian legend. Rand's taking of Callandor from the Heart of the Stone of Tear to declare himself is a typical if creative 'Sword in the Stone' story; the connection becomes even more clear when he drives the sword into the stone floor of the Heart before leaving Tear for Rhuidean.
In the aftermath of the Cleansing of Saidin. it is very clear that the sa'angreal was in Cadsuane's possession. From her POV she states that the access key for the male Choedan Kal and Callandor were in her saddle bags.