Heat Vision is a superpower possessed by Kryptonians and Daxamites when exposed to the yellow sun; see Photonucleic Effect. It is the ability to release massive amounts of solar energy stored within the body through the eyes, in focused beams of heat (Presumably, as infra-red light).
It was originally considered to be a subset of X-Ray Vision; for example, Superman was known to melt metal with "The heat of his X-Ray vision". As his powers developed and became more defined, however, Heat Vision became distinguished as an individual power. The temperature of heat vision or a laser beam is stated to be 5000 degrees (presumably Celsius), making it capable of burning through the toughest steel, material and armor.
When angry, or to make a threat, Kryptonians may cause their eyes to glow a threatening red, without any strong beams actually emerging from their eyes.
Martians - Martians can shoot fire/beams from their eyes.
Kryptonians and Daxamites - Kryptonians and Daxamites can shoot fire/beams from their eyes.
In most depictions, Heat Vision has been shown as a solid red beam, as you see in the image to the right. However, in Smallville and Superman Returns, heat vision was pictured as beams of heat distortions in the air—that is, the beams themselves are invisible, but their intense heat causes the air to ripple and shimmer.
In other works, "invisible" heat vision means that the beams are relatively low power and is sometimes used by Superman in a discreet manner as Clark Kent. In the later seasons of Smallville, the beams become solid orange, implying that the appearance changed because Clark has become more powerful as he ages.
In the Supergirl series, it's blue.
One variation on this power is known as Flame Vision. Held by an angelic incarnation of Supergirl (Linda Danvers), this vision was not pure heat waves but actual flames which would erupt from her eyes. The Smallville version of Bizarro also has heat vision that looks more like flames.