Analysis of Covariance (ANCOVA):
Analysis of covariance is a more sophisticated method of analysis of variance. It is based on inclusion of supplementary variables (covariates) into the model. This lets you account for inter-group variation associated not with the "treatment" itself, but with covariate(s).
Suppose you analyze the results of a clinical trial of three types of treatment of a disease - "Placebo", "Drug 1", and "Drug 2". The results are three sets of survival times, corresponding to patients from the three treatment groups. The question of interest is whether there is a difference between the three types of treatment in the average survival time.
You might use analysis of variance to answer this question. But, if you have supplementary information, for example, each patient´s age, then analysis of covariance allows you to adjust the treatment effect (survival time, in this case) to a particular age, say, the mean age of all patients. Age in this case is a "covariate" - it is not related to treatment, but can affect the survival time. This adjustment allows you to reduce the observed variation between the three groups caused not by the treatment itself but by variation of age.
If the covariate(s) are associated with the treatment effect, then analysis of covariance may have more power than analysis of variance.