I) The Casimir invariants of a Lie algebra $L$ over a field $\mathbb{F}$ are the central elements of the universal enveloping algebra $U(L)$.
Example: The angular momentum square $\vec{J}^2$ is a quadratic Casimir invariant of the Lie algebra $L=sl(2,\mathbb{C})$.
II) Given a bilinear associative/invariant form $B:L\times L\to \mathbb{F}$, one can create a quadratic Casimir invariant, as explained on this Wikipedia page.
A simple Lie algebra has a unique bilinear associative/invariant form (up to an overall normalization factor), namely the Killing form.
As a consequence, a simple Lie algebra has a unique quadratic Casimir invariant (up to an overall normalization factor)
$$C_2 ~:=~ t_a \otimes t^a~\in~ U(L), \qquad t^a~:=~(\kappa^{-1})^{ab} t_b, \qquad \kappa_{ab}~:=~{\rm tr}({\rm ad} t_a\circ{\rm ad} t_b). $$
III) More generally, a semisimple Lie algebra that is built from $m$ simple Lie algebras has
a basis a $m$ quadratic Casimir invariant.
Example: The linear combination
$$\alpha_L \vec{J}_{\!L}^2+\alpha_R \vec{J}_{\!R}^2$$
is a quadratic Casimir invariant of the Lie algebra $L=sl(2,\mathbb{C})_L\oplus sl(2,\mathbb{C})_R$ for arbitrary constants $\alpha_L,\alpha_R\in\mathbb{C}$.
IV) There also exist cubic and higher-order Casimir invariants. For a semisimple Lie algebra $L$, e.g.,
$$C_n ~:=~ {\rm str}({\rm ad} t_{a_1}\circ\ldots\circ{\rm ad} t_{a_n}) t^{a_1} \otimes\ldots\otimes t^{a_n}~\in~ U(L),$$
where ${\rm str}$ denotes symmetrized trace. They are not all independent, though.
V) Finally, in response to Art Brown's comment: Racah's theorem states that the number of independent Casimir invariants for a complex semisimple Lie algebra $L$ is equal to the rank of the Lie algebra $L$.
There exist generalizations of Racah's theorem to non-semisimple Lie algebras, see e.g. B.G. Wybourne, Classical Groups for Physicists, 1974, p. 142.