You can classify mesons via two questions:
How do they transform under P and C?
$P$ is the eigenvalue under parity transformations and $C$ is the eigenvalue under charge conjugation. They can take on the values $\pm 1$, so the possibilities are $$PC = \big\{(++), \, (+-),\,(-+),\, (--)\big\}$$
What is their quark structure?
Here I'm talking about constituent quarks, that is no quark-antiquark pairs that may appear inside a meson.
If you answer those two questions, you can find an associated name in tables like this one, which is a good summary of all the different mesons.
Additionally, you can ask:
What is their total angular momentum?
As you can see in the review I linked above, the lowest possibilities for the total angular momentum $J$ are $$ J^{PC} = 0^{++},\, 1^{+-},\, 0^{-+},\, 1^{--}$$
They are the ground state, whereas higher-$J$ represent excited states.