Principal Components and other forms of factor extraction methods
There are several different types of analyses which attempt to identify clusters of correlated variables (factors) which comprise multivariate data. One of the most common ways of extracting these factors being principal components analysis (PCA).
PCA is an exploratory method used when the primary concern is parsimony and seeing how few clusters (linear combinations of the observed variables called factors) are needed to account for the correlations between the variables.
PCA is also used when you suspect most of the variance in the variables can be accounted for by the other variables. ie the variables have small unique variances.
Other ways of extracting factors, including principal axis factoring (PAF), also called common factor analysis, are used when you are interested in identifying variable clusters by seeing which variables are related to the dimensions (eg by examining factor loadings). It is also recommended if you know little about how much variance in the variables is unexplained by the other variables in the analysis.
These other (non-PCA) extracting methods may also be further used when you have a good idea of the make-up of these variable clusters via confirmatory factor analysis in structural equation modeling.
So to summarise, PCA identifies the number of variable clusters and whilst non-PCA methods are useful for refining this by defining these clusters.
In most cases PCA and non-PCA methods will yield similar results.
Reference
Hair Jr, JF, Anderson, RE, Tatham RL, Black WC (1998) Multivariate Data Analysis Fifthe Edition. Prentice-Hall.
(there is also a 2005 6th Edition).