Multiple correlation analysis often requires the control for a third variable, so you are asking: What is the relationship between Var1 and Var2 after allowing for their relationship with Var3. In that case, the relationship between Var1 and Var2 would be a "partial correlation."
Whenever we are interested in studying the joint effect of two or more variables on a single variable, multiple correlation gives the solution of our problem. In fact, multiple correlation is the study of combined influence of two or more variables on a single variable.