Lyotropic liquid crystals result when amphiphiles, which are both hydrophobic and hydrophilic, dissolve into a solution that behaves both like a liquid and a solid crystal. This liquid crystalline mesophase includes everyday mixtures like soap and water. 
To break the word down, "lyo" and "tropic" mean, respectively, "dissolve" and "change." Historically, the term was used to describe the common behavior of materials composed of amphiphilic molecules upon the addition of a solvent. Such molecules comprise a water-loving hydrophilic head-group (which may be ionic or non-ionic) attached to a water-hating, hydrophobic group.
The micro-phase segregation of two incompatible components on a nanometer scale results in different type of solvent-induced extended anisotropic arrangement, depending on the volume balances between the hydrophilic part and hydrophobic part. In turn, they generate the long-range order of the phases, with the solvent molecules filling the space around the compounds to provide fluidity to the system.
In contrast to thermotropic liquid crystals, lyotropic liquid crystals have therefore an additional degree of freedom, that is the concentration that enables them to induce a variety of different phases. As the concentration of amphiphilic molecules is increased, several different type of lyotropic liquid crystal structures occur in solution. Each of these different types has a different extent of molecular ordering within the solvent matrix, from spherical micelles to larger cylinders, aligned cylinders and even bilayered and multiwalled aggregates.