The concept of nonorthogonal localized molecular orbital (NOLMO) is investigated in this paper. Given a set of the commonly used canonical molecular orbitals, a direct minimization algorithm is proposed to obtain both the orthogonal localized molecular orbitals (OLMO) and NOLMO by using the Boys criterion and conjugate gradient minimization. To avoid the multiple-minimum problem, the absolute energy minimization principle of Yang is employed to obtain initial guesses. Contrary to the early conclusion drawn by Lipscomb and co-workers who claimed that OLMOs and the corresponding NOLMOs are more or less the same, we found that NOLMOs are about 10%-30% more localized than OLMOs. More importantly, the so-called "delocalization tail" that plagues OLMOs is not present in NOLMOs, showing that NOLMOs are more compact and less oscillatory and capable of providing greater transferability in describing the electronic structure of molecules. We also found that main lobes of NOLMOs are slightly larger in size than those of OLMOs because of the normalization requirement. These features establish NOLMOs to be valuable as building blocks in electronic structure theory and for the understanding of chemical bonding. They show the promise for the utilization of NOLMOs-the most localized possible-in the linear scaling approaches of the electronic structure theory for molecules and solids.