Factor analysis, technically, is an attempt to recover the correlation matrix of each variable with all other variables. One starts with a data matrix with a score for each participant on each variable. One then finds the correlation matrix, or correlation of each variable with every other variable. By submitting the correlation matrix to a standard computer program, one finds the factor matrix, which is made up of a correlation of each variable with each factor. By turning this factor matrix on its side one derives the factor matrix transpose, and by multiplying the factor matrix by the factor matrix transpose, one arrives at an approximation matrix comprised of the correlation of each variable with every other variable. The approximation matrix is approximately equal to the correlation matrix. The object of factor analysis is to recover the correlation matrix using a few factors. While some information is lost in the process, hopefully not very much information is lost.
Once one obtains a factor matrix (comprised of correlations of items with factors), it is possible to rotate the factor matrix. Although there are many methods of rotation--actually, infinitely many--the method most favored by psychologists is rotation to simple structure. Simple structure means that each item has a high loading on (or correlation with) one and only one factor. Simple structure is intended to improve the interpretability of factors by including in each factor only items that are good measures of that factor.
Factor analysis is the basis for such trait approaches as the five-factor model and general intelligence. Indeed, factor analysis is probably the most widely used method in personality psychology.