Liquid crystals are materials that exhibit phases of matter between that of liquid and solid.
Seriously. That’s it.
To be more technical: liquid crystals exhibit phases that do not possess long-range 3D positional order. All liquid crystals exhibit some amount of disorder, be it in one dimension, or more. Liquid crystals also possess “orientational order” which means the molecules exhibit (on average) coherent direction of their molecular long axis (and sometimes short axis). There is, therefore, a boggling number of phases that can exist between crystalline solids and fully isotropic liquids.
Liquid crystals (mesogens) come in two distinct types: thermotropic and lyotropic.
Thermotropic: thermotropic liquid crystals exhibit phase behavior based on temperature. The higher the temperature, the more disordered the phase. Phases range from nearly crystalline to nearly isotropic. These are most commonly found in things like thermometer stickers, thin-film displays, and those “hidden” stickers that you rubbed to see the picture underneath (see: 80s-era transformers).
Lyotropic: lyotropic liquid crystalls exhibit phase behavior based on concentration of components in a mixture. Many soaps and fatty acids function as lyotropic liquid crystals, and indeed, many of the biomolecules in our body have the capability of acting as liquid crystals (even DNA!).
Here are a variety of resources you can use to learn about the basics of liquid crystals:
The Wikipedia entry on liquid crystals
A short tutorial on liquid crystals at Case Western Reserve University
A great online resource from Bohan Senyuk on liquid crystals
The topic page at NobelPrize.org