This paper describes the significance interpretation, comparison to other molecular descriptors, and QSPR/QSAR applications of a new set of molecular descriptors: atom, atom type, and total molecular quadratic indices. The features of the kth total and local quadratic indices are illustrated by examples of various types of molecular structures, including chain lengthening, branching, heteroatoms content, and multiple bonds. The linear independence of the local (atom type) quadratic indices to others 0D, 1D, 2D, and 3D molecular descriptors is demonstrated by using principal component analysis for 42 heterogeneous molecules. It is concluded that the local quadratic indices are independent indices containing important structural information to be used in QSPR/QSAR and drug design studies. In this sense, molecular quadratic indices were used to the description and prediction of the boiling point of 28 alkyl alcohols and to the modeling of the partition coefficient (log P), specific rate constant (log k), and antibacterial activity of 2-furylethylene derivatives. These models were statistically significant and showed very good stability to data variation in leave-one-out (LOO) cross-validation experiment. The comparison with the other approaches also revealed good behaviors of our method in this QSAR study.