I have read may places and take that it is customary to use continuous covariates in Analysis of Covariance. I am curious to know if it is acceptable and possible to use a categorical variable as a covariate. Can anyone help?
Yes, you may use a categorical covariate. However, if it has more than two levels, you will need to re-express the categories into C - 1 dummy variates (where C is the number of categories you have in your data set). If only two levels, then coding them as "1" and "2", respectively, will work just fine.
An alternative to this design is to treat the categorical variable as a blocking factor. Doing so allows you still to separate the influence of that variable from other factor(s) in your design, as well as investigate potential interactions.
Another alternative is treat your problem as a regression analysis, which is algebraically equivalent to ANCOVA. In that case, you include the categorical variable in the regression as a set of dummy variables, as described by David Morse.