Most liquids freeze by crystallization, formation of crystalline solid from the uniform liquid. This is a first-order thermodynamic phase transition, which means that as long as solid and liquid coexist, the temperature of the whole system remains very nearly equal to the melting point due to the slow removal of heat when in contact with air, which is a poor heat conductor. Because of the latent heat of fusion, the freezing is greatly slowed and the temperature will not drop anymore once the freezing starts but will continue dropping once it finishes. 
Crystallization consists of two major events, nucleation and crystal growth. "Nucleation" is the step wherein the molecules start to gather into clusters, on the nanometer scale, arranging in a defined and periodic manner that defines the crystal structure. "Crystal growth" is the subsequent growth of the nuclei that succeed in achieving the critical cluster size.