Term for a matching which is perfect on one side only
Such a matching is said to saturate one of the sides.  More specifically, if $M$ is a matching of a graph $G$ and $v$ is incident to one of the edges of $M$, we say that $M$ saturates $v$.  If $A$ is a set of vertices, then we say that $M$ saturates $A$ if every vertex in $A$ is saturated by $M$.

Specifically for your case, the matching saturates the smaller of the two sides of the bipartition.