Distillation is the process of purification whereby a solution is sucessively evaporated and condensed to extract pure water from the heavier solids in the solution. In practical terms, H2O- which is lighter - is evaporated, condensed, and collected while the heavier solids are left behind. Distillation machines are not very practical for common applications, and hence you don't see any available on the market. Some purified, market-bought bottled waters are distilled.
R/O forces a solution through a semi-permeable membrane, "sifting" out larger molecules to form purified water. Although the membrane does not reject 100% of unwanted molecules, the water produced is, for all intensive purposes, pure enough for most consumers. Because of it's small size and low cost, R/O is the most practical water purification method for the average person.
Deionizatoin uses charged ions to strip minerals in a solution to purify it (often called softening the water since it removes primarily carbonates and earth metals). When used in conjunction with R/O, deionization can further purify the water beyond what R/O can do alone.