The Rankin-Selberg convolution is a process that creates a new L-function from a pair of old ones.
Given a pair with Selberg data \bigg(d_1,Q_1,\bigg\{\mu_{1,i}\bigg\},\epsilon_1\bigg) and \bigg(d_2,Q_2,\bigg\{\mu_{2,i}\bigg\},\epsilon_2\bigg), there exists another L-function with Selberg data
\bigg(d_1+d_2,Q,\bigg\{\mu_i \bigg\},\epsilon\bigg),
where the Q divides Q_1\cdot Q_2, and the relations for the \mu's and \epsilon's are not clear (to me). In the case of Artin L-functions (which conjecturally, cover all Langlands L-functions), if the two forms are
L(s,\pi_1) = \prod_i^n (1-p^{-s} \alpha_i(p))
and L(s,\pi_2) = \prod_i^n (1-p^{-s} \beta_i(p)),
then L(s,\pi_1 \times \pi_2) = \prod_p \prod_{i,j}^n (1-\alpha_i(p) \beta_j(p) p^{-s})^{-1}.
Observe that this new L-functions has n^2 Satake-Langlands parameters. This process is used for computing symmetric powers and exterior products of L-functions.