Just to set the terminology, a closed geodesic is called simple when it is a smooth embedded circle in the Riemannian or Finsler manifold. If a closed Riemannian manifold is not simply connected one can easily show that the shortest non-contractible curve is a simple closed geodesic.
Existence and multiplicity results for simple closed geodesics on simply connected manifolds are currently available only for 2-dimensional manifolds. Indeed, as it has been already mentioned, in higher dimension being simple is a generic condition: given a closed geodesic $\gamma$, with a tiny perturbation of the metric one can produce a simple closed geodesic close to $\gamma$ for the new metric. For this reason, it is not obvious how to obtain a simple closed geodesic in higher dimension by means of variational methods: a generic metric has at least one (and often more), but a very unlucky metric in principle might have none.

The existence of closed geodesics possibly with self-intersections was proved by Birkhoff for Riemannian 2-spheres, and later by Lusternik-Fet in full generality (as it was mentioned above by Rivin and Alvarez Paiva). These proofs remain valid for general Finsler metrics.

Beside the already mentioned case of Riemannian $S^2$, the existence of a second closed geodesic is already hard business: for suitably non-degenerate Riemannian metrics, it was a theorem of Fet from the 1960s. The theorem should hold as well for reversible Finsler metrics (i.e. Finsler metrics $F$ with the property that $F(v)=F(-v)$ for all vectors $v$). For general, non-reversible, Finsler metrics on $S^2$ it was a theorem of Bangert-Long from the late 2000s. For higher dimensional simply connected closed manifolds equipped with non-degenerate and possibly non-reversible Finsler metric, the existence of a second geodesic is a recent theorem of Duan-Long-Wang (JDG, 2015).

For simply connected closed manifolds whose cohomology ring $H^*(M;\mathbb Q)$ is not generated by one element, a celebrated result of Gromoll-Meyer from 1971 implies that there are always infinitely many Riemannian or Finsler closed geodesics (possibly with self-intersections). The only simply connected manifolds not covered by this theorem are those who have the rational cohomology of a compact rank-one symmetric space, that is, the rational cohomology of $S^n$, $\mathbb RP^n$, $\mathbb CP^n$, $\mathbb HP^n$, or $\mathrm{Ca}P^2$. The case of $S^2$ was settled in a combination of celebrated papers by Bangert, Franks, and Hingston. For the other CROSSes, Hingston and Rademacher proved that a suitably generic Riemannian metric has infinitely many closed geodesics.

On non-simply connected closed Riemannian or Finsler manifolds, it is very easy to find infinitely many closed geodesics when, for instance, the fundamental group is abelian and has rank larger than 1. The statement is still true, but non-trivial, if the fundamental group is only assumed to be infinite abelian (the most difficult case being when it is $\mathbb Z$); it was proved by Bangert-Hingston in the 1980s.

In his book "Lectures on closed geodesics" from the 1980s, Klingenberg has a proof of the existence of infinitely many closed geodesics for any closed Riemannian manifold (of dimension at least two). However, the proof was crucially based on a divisibility lemma that was later found to be wrong. There are unfortunately other mistakes in the literature, notably by Alber (his so-called Alber Lemma) and Klingenberg. The results that I mentioned above are universally accepted and have a solid proof, but one should be careful with older results in the published literature which claim stronger conclusions.

The unconditional existence of infinitely many closed geodesics, or even of a second closed geodesic on a general closed Riemannian manifold of arbitrary dimension, are still open problems.

The above are among the most significant results in the field, but I certainly forgot to mention many others.