Factor analysis is a set of techniques designed to find latent variables in sets of data.  A latent variable is one that cannot be directly measured.  For example, height is not latent, because it can be directly measured.  But many abilities, traits and beliefs are latent.  These include such things as intelligence, political beliefs, depression, and so on.

To measure any of these, we would typically ask many questions that we thought were related to the latent trait, and then factor analyze them to determine the best way to score them, whether all the questions were related to the trait, how many traits there were, and related questions.

There are many types of factor analysis, but they fall into two large groups:  Exploratory and confirmatory.  In exploratory factor analysis we have no preconceived notions of the factor structure, but in confirmatory we attempt to replicate some earlier results.

There are two phases to factor analysis: extraction of the factors and rotation, and there are a variety of methods for each.   The rotations can be divided into orthogonal and oblique methods.  In orthogonal factor analysis, we require that each trait (or factor) is orthogonal (uncorrelated) with the other traits.  In oblique factor analysis, we do not make this assumption.