A state of matter, in the sense that solid, liquid and plasma are states of matter. Bose-Einstein condensates form from matter that has been cooled to near absolute zero. Predicted in the 1920s by Satyendra Nath Bose and Albert Einstein based on Bose's work on rules for deciding when two photons should be counted up as either identical or different. Einstein applied Bose's theories to atoms instead of just photons, and discovered that at low enough temperatures it became difficult to consider individual atoms to be separate particles anymore.
The effect can be understood in broad outline by considering the Uncertainty Principle which states, roughly, that it is impossible to know both a particle's velocity and a particle's position simultaneously with certainty. When a group of atoms is cooled to a low enough temperature, however, their velocities become very certain; they must be moving very slowly, or stated more technically they must have low quantum energy levels. This causes their positions to "smear out," effectively causing the individual atoms to overlap each other. In a Bose-Einstein condensate, the many overlapping atoms can be considered to be a single super-atom, with all of its constituent atoms sharing a single quantum state and a single location in space.
A Bose-Einstein condensate was not actually created in a lab until June 5 1995, when Eric Cornell and Carl Wieman used a combination of laser cooling (a technique that Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips won the 1997 Nobel Prize for Physics for inventing) and magnetic evaporative cooling to cool a cloud of approximately 2000 rubidium ions to 20 billionths of a degree above absolute zero, the lowest temperature ever achieved at that time. This was cold enough to form a Bose-Einstein condensate .
Bose-Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, causing them to break back down into individual atoms again; it will likely be some time before any practical applications are developed for them. However, several interesting properties have already been observed in experiments. Bose-Einstein condensates have extremely high optical densities, resulting in extremely low measured speed of light with in them; some condensates have slowed beams of light down to mere meters per second, slower than a human can move on a bicycle. A rotating Bose-Einstein condensate could be used as a model black hole, allowing light to enter but not to escape. Condensates could also be used to "freeze" pulses of light, to be released again when the condensate breaks down. Research in this field is still young and ongoing.