Factor: A source of variance in the response. A categorical factor is measured in categorical levels, whereas a covariate factor is measured on a scale of continuous (or sometimes ordinal) variation. A statistical model might be constructed to test the influence of a factor as the sole explanation (Y = A + ε) or as one of many factors variously crossed with each other or nested within each other.
A fixed factor has levels that are fixed by the design and could be repeated without error in another investigation. The factor has a significant effect if sample means differ by considerably more than the background variation, or for a covariate, if the variation of the regression line from horizontal greatly exceeds the variation of data points from the line.
A random factor has levels that sample at random from a defined population. A random factor will be assumed to have a normal distribution of sample means, and homogenous variance of means, if its MS is the error variance for estimating other effects (e.g., in nested designs). The random factor has a significant effect if the variance among its levels is considerably greater than zero.
Doncaster, C. P. & Davey, A. J. H. (2007) Analysis of Variance and Covariance: How to Choose and Construct Models for the Life Sciences. Cambridge: Cambridge University Press.