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The statement is true. If $M$ is generated by $m_1,\dots ,m_r$, there is a surjection $R^r\twoheadrightarrow M$. The right exactness of the tensor product of modules gives a surjectionA finite direct sum of Noetherian modules is Noetherian, and a quotient of a Noetherian module is Noetherian. Hence $M{\otimes }_{R}N$ is Noetherian.

An $R$-module $M$ is Noetherian when every submodule is finitely generated, equivalently when every ascending chain of submodules stabilizes.

A free module is an $R$-module with a basis: every element has a unique expression as a finite linear combination of basis elements.

A flat module $M$ is one for which the tensor functor $-{\otimes }_{R}M$ is exact. Since tensor products are always right exact, it is equivalent to require that tensoring with $M$ preserve injections.

A projective module $P$ has the lifting property: for every surjection $E\twoheadrightarrow F$ and every map $P\to F$, there is a map $P\to E$ making the resulting triangle commute. Equivalently, $P$ is a direct summand of a free module.