Generally, Covariance is computed assuming that two random variables are independent. But in a situation that warrants some association between these two variables, how can we compute covariance.
A covariate is a variable that potentially varies along with the variable of primary interest.
Covariance is a measure of the relationship between two variables (if divided by the standard deviations it becomes the correlation).
Covariance is generally a single number (though you can have a covariance matrix where each value is a covariance between two variables). A covariate is generally a vector with one observation per subject/unit.
You can calculate the covariance between two covariates.