Rotation may simplify the factor structure by limiting the number of large correlations that variables have with certain factors. Ideally you want each variable to load significantly onto one factor. Rotation can help achieve this and thus help differentiate the factors from one another.
There are two basic types of rotation, orthogonal and oblique. You should always start with an orthogonal rotation like Varimax. Varimax keeps the factor axes at right angles and assumes that they are independent or uncorrelated. Other less common orthogonal techniques in your list are quartimax and equamax.
Oblique methods in your list are Direct Oblimin and Promax. Oblique methods allow the factors to be correlated. I tend to use Direct Oblimin when I have reason to suspect that the factors are related. Also SPSS will produce a factor correlation matrix. If the factors are highly correlated you may want to use an oblique rotation method.
You should experiment with both orthogonal and oblique rotation methods. Choose a rotation method that produces dimensions which are most consistent with your research expectations.