I'll add something to the previous answer. The first way to construct multilinear maps is pretty recent and was introduced by Sanjam Garg, Craig Gentry and Shai Halevi. What we want is given groups $G_1,\ldots,G_n$ and $G_T$ a map:
$$e:G_1\times\cdots\times G_n\to G_T$$
that satisfies the linearity property in DrLecter's answer. It's worth nothing here, that $G_1,\ldots,G_n$ do not necessarily have to be distinct groups. Often, they would be the same group and we would call this a symmetric $n$-linear map.
Current constructions are typically called leveled multilinear maps. In the symmetric case this can be described as follows. Assume that you have groups $G_1,\ldots,G_n$ and bilinear maps $e_{i,j}:G_i\times G_j\to G_{i+j}$ for all $i,j > 0$ that satisfy $i+j\leq k$. We can construct a symmetric $n$-linear map $e:G_1\times\cdots\times G_1\to G_n$ from this by recursively defining:
$$e_2 = e_{1,1},\;\; e_n(g_1,\ldots,g_n) = e_{1,k-1}(g_1,e_{n-1}(g_2,\ldots,g_n))$$
For example if $n=3$, then we would compute:
$$e_3(g_1^{a_1},g_2^{a_2},g_3^{a_3}) = e_{1,2}(g_1^{a_1},e_{1,1}(g_2^{a_2},g_3^{a_3}))=e_{1,2}(g_1,e_{1,1}(g_2,g_3)^{a_2a_3})^{a_1}=e_{1,2}(g_1,e_{1,1}(g_2,g_3))^{a_1a_2a_3},$$
which shows that $e_3$ is $3$-linear. The asymmetric case is slightly more complicated (it's a bit heavy notation wise and the subscripts become sets instead of integers). Thus, current constructions have a bit more structure than a pure $n$-linear group. This is both good and bad in that more versatile structures can allow for more elaborate constructions, but on the other hand, if we only need $n$-linearity, then the extra structure might lead to possible attacks. However, in a generic group setting the leveled $n$-linear and $n$-linear settings are essentially equivalent, so there might not be that much danger.