http://en.wikipedia.org/wiki/CovariateIn statistics, a covariate is a variable that is possibly predictive of the outcome under study. A covariate may be of direct interest or it may be a confounding or interacting variable.
One could consider an example of weight vs. height.
Weight could be plotted vs. height for 100 subjects, and a regression analysis done.
In this case we would call height the independent variable (or explanatory variable), and weight (the response variable). Height would be a covariate.
However, a covariate could also mean another source for the variation in the weight (e.g. one could consider eating habits, or level of exercise). One that is not measured or which is actually confounding.
In terms of factor the word factor is used in ANOVA.
A single factor ANOVA could be an experiment in which 5 different dosage levels of an anti-depressant are each tested on 10 different patients.
It is often written in an ANOVA table (which in this case would have 5 rows and 10 columns).
A two-factor ANOVA could be an experiment in which 5 different dosage levels of an anti-depressant and 5 different kinds of exercise routines are used.
In this two-factor ANOVA you could see whether certain exercise routines and anti-depressants work with each other, against each other, or have no interaction effect.
If you had 50 patients you could place 2 patients into each of the 25 different combinations of dosage levels and exercise routines.