A cosmic string is a line in space that is a discontinuity of the universe's typical near-Euclidean geometry of space-time. Theories were developed in the 1970s about their formation in the early universe, their explaining power regarding some observed phenomena (specifically the clumping of matter), and possible methods of detecting or verifying their existence. Alternate explanations have become more popular than their explaining power, apparently fitting nicely with some observations, and there has been no widely-recognized sign of their existence.
They are not the eponymous strings of string theory, though string theory provides some theory behind them. They are (theorized) "defects" in geometry as is a black hole or wormhole. A circle encircling a cosmic string would have less than 2π circumference. They would (effectively) show mass, propagating from a line. Gravitational lensing by one would create two full images, e.g., two views of the same galaxy.
An activity occurring near the beginning of the universe, symmetry breaking is considered a possible source of cosmic string (one-dimensional) discontinuities, and also a possible source of other kinds of such discontinuities (topological defects), magnetic monopoles (zero-dimensional), domain walls (two-dimensional), and textures (three-dimensional).