Degrees of freedom (d.f.) are the total number of independent pieces of information contributing to the component of variation, minus the number of pieces required to measure it. Analysis of variance is always reported with two values of degrees of freedom. The first informs on the number of test samples, and the second informs on the number of independent and random replicates available for calibrating the test effect against the background 'error' variation.
For example, a result F2,12 = 3.98, P < 0.05 indicates a significant effect at a threshold Type-I error of α = 0.05. This outcome applies to a design with a = 3 samples or treatment levels, giving 2 test d.f. (= a - 1, since one grand mean is required to test variation of a sample means). The a samples were allocated amongst a total of N = 15 sampling units, giving 12 error d.f. (= N - a, since a sample means are required to test within-sample variation of N observations).
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.