Degrees of Freedom:
For a set of data points in a given situation (e.g. with mean or other parameter specified, or not), degrees of freedom is the minimal number of values which should be specified to determine all the data points.
For example, if you have a sample of N random values, there are N degrees of freedom (you cannot determine the Nth random value even if you know N-1 other values). If your data have been obtained by subtracting the sample mean from each data point (thus making the new sample mean equal to zero), there are only N-1 degrees of freedom. This is because if you know N-1 data points, you may find the remaining (Nth) point - it is just the sum of the N-1 values with the negative sign. This is another way of saying that if you have N data points and you know the sample mean, you have N-1 degrees of freedom.
Another example is a 2x2 table; it generally has 4 degrees of freedom - each of the 4 cells can contain any number. If row and column marginal totals are specified, there is only 1 degree of freedom: if you know the number in a cell, you may calculate the remaining 3 numbers from the known number and the marginal totals.
Degrees of freedom are often used to characterize various distributions. See, for example, chi-square distribution, t-distribution, F distribution.