When the molecules in a liquid line up into a very regular pattern (a crystal), we say that they freeze. This type of process happens at a sharply defined freezing temperature. Vegetable oil consists of molecules in long chains called lipids, usually with a mixture of molecules of different sizes and shapes. That makes it hard for the oil to form a regular crystal, suppressing ordinary freezing. There are exceptions, however, such as cocoa butter, which freezes at 34 °C.
Cooling down oil does make it resist flow however, becoming more "viscous". Eventually, if you get it cold enough, the oil will seem to be quite solid, as the molecules lose the energy needed to move around. In that sense, you can say that oil does in fact freeze, but it has no sharply defined freezing point, like water. Glass is similar, in that it gets soft and eventually quite fluid as it is heated, but there is no one sharp melting temperature. The molecules get stuck when cold, but not in any simple regular arrangement.
Tom and Mike
(republished on 07/25/06)