Yes, it really depends on the statistical test you are interested in using. If you wish to test whether the probability of getting a head or a tail is equal (i.e., P(H)=P(T)=0.25 for two coins), then you may want to use the Chi-square goodness of fit test.
The degree of freedom for this test, then, is Number of Characteristics - 1 = 2 - 1 = 1. Notice there are two characteristics here - tail and head.
Feel free to correct me if you think otherwise.