I'd like to understand the structure of the free loop space of $S^n$ for small values of $n$. Here "understand" means roughly that I'd like to know a CW complex with the same homotopy type.
I already "understand" the pointed loop space of $S^n$ using the James reduced product. (See Hatcher, Algebraic Topology, Section 4.J.) Is there something analogous for the free loop space? Alternatively, is there some simple relationship between the free loop space and the pointed loop space?
Stably the free loop space of the suspension of a connected space splits up, just as the based loop space does. Just as $\Omega\Sigma X$ is stably the wedge of the smash products $X^{\wedge n}$, $L\Sigma X$ is stably the wedge of $S^1_+\wedge_{C_n}X^{\wedge n}$. Here $C_n$ is a cyclic group of order $n$ acting freely on $S^1$ and permuting the factors in the smash product. This is even correct $S^1$-equivariantly in a weak sense.
EDIT Here is an answer more directly relevant to the question:
Suppose $X\sim BG$. Even if the group $G$ is not discrete, we still have the equivalence that Qiaochu mentioned in a comment, between $LX$ and the homotopy orbit space for the conjugation action of $G$ on $G$. And this in turn is equivalent to the two-sided bar construction for $G$ acting on itself on both sides, i.e. the cyclic bar construction $N^{cyc}G$. And if we have another topological monoid $M$ equivalent to $G$, for example the Moore loops on $X$, or the James construction $JY$ if $X=\Sigma Y$, then $N^{cyc}M$ is another model for $LX$.
Furthermore, there is a nice way of thinking of (the realization of the simplicial space) $N^{cyc}M$, which informally goes like this: A point is given by a finite subset $T$ of $S^1$ together with a labeling: a function $T\to M$. The set $T$ and the labels can move. If $T$ moves in such a way that several points come together, the label of the new point is the product of the old labels. If a label becomes $1\in M$ then the point may be deleted from $T$.
This is even equivariantly correct regarding the $S^1$-action, in some sense (not good for fixed-point spaces of the whole group, but OK for finite subgroups).
When $X=S^{n}$ and $M=JS^{n-1}$, this gives a pretty good equivariant cell structure for $LS^n$.
The free loop space $LM$ forms a fibre bundle over $M$, by evaluating a loop at $1$, with fibre $\Omega M$. So, in theory, one "knows" a lot if one knows the twisting involved when building this bundle/fibration. There are several computational tools that help you to compute Betti numbers of $LM$ etc (rational homotopy theory, Hochschild homology etc.) but since you ask for CW complexes for small values of $n$, here is an attempt :
For any group $G$, it is clear that $LG\cong G\times\Omega G$.This is not an isomorphism as groups! In particular, when $n=1$ or $3$, we have a description of $LS^1\sim S^1\times\mathbb{Z}$ and $LS^3$. Even spheres and higher odd dimensional spheres ($5$, $9$ and higher) do not fall in this list. The Betti numbers of $LS^n$ are computable and these are bounded (either $0$, $1$ or $2$). This should give you a (rational) model for the $n$th skeleton of $LM$. And, in a non-rigorous sense which is meaningful in the context of algebraic structures, $LS^\textrm{odd}$ "behaves" like $S^\textrm{odd}\times\Omega S^\textrm{odd}$.
There is a relationship between $\Omega M$ and $LM$ via homological algebra for simply connected manifolds $M$. The Hochschild homology of the chains on $\Omega M$ is isomorphic to the cohomology ring of $LM$. This is perhaps as precise as it gets; any simple relationship that you may want needs to account for the fact that $LM$ has a circle action while $\Omega M$ doesn't.
The next best thing to the knowledge of a CW-structure is probably the knowledge of the integral homology (not just the Betti numbers). Perhaps Ziller was the first to compute it in The Free Loop Space of Global Symmetric Spaces. Note the following:
In the case of $n>2$, where $LS^n$ is simply-connected, this gives the number of cells of a minimal CW-structure (see Hatcher's Algebraic Topology).
The behavior in the case $n$ even and $n$ odd is very different. The difference might be seen as a differential in a Serre spectral sequence, but I like to view it as coming from the hairy ball theorem that we cannot find a non-vanishing vector field on an even-dimensional sphere.
The latter connection becomes clear if we look at explicit generators of the integral homology. I have done this in Section 5.2.1 of my paper Spectral Sequences in String Topology (I apologize for the self-advertisement).
I recently learned the following description of the topology of the loop space at the European Talbot on free loop spaces, and is probably what Ziller does in the paper in the answer of Lennart Meier.
The idea is to use Morse theory for the standard round sphere. The critical points of the energy functional $E:LS^n\rightarrow \mathbb R$ are the closed geodesics.
What are the closed geodesics? The trivial ones are the constant loops. The collection of all constant loops forms a family diffeomorphic to $S^n$. Other closed geodesics are great circles. The family of great circles can be described as $T_rS^n$ where $r$ is the energy of a prime great circle. But there are more closed geodesiscs. These are the iterates of the great circles. The iterates form families $T_{k^2r}S^n$, which are of course all diffeomorphic to the unit sphere bundle of $S^n$.
The wonderful luck is that $E$ is Morse-Bott and satisfies Palais-Smale. That means that you can build up $LS^n$ by attaching disc bundles along the critical sets. Thus we see that the homotopy type is $$ LS^n\simeq S^n\cup D_{\lambda(1)}T_{r}S^n\cup D_{\lambda(2)}T_{2^2r}S^n\cup\ldots $$ In the union we make some identifications on the boundary of the disc bundles which I do not want to specify here. The number $\lambda(k)$ is the Morse-Bott index of the critical set $T_{k^2r}S^n$ (If I remember correctly it is $(2k-1)(n-1)$).
One can show that the attachment of the discbundles is homological trivial in this situation. Thus the homology of the loop space the sum of the homology of the sphere and copies of the homology of the unit tangent bundle of the sphere shifted in dimension.
The unit tangent bundle of the sphere has two torsion if $n$ is even, but not when $n$ is odd. This explains the difference in the homology of the free loop space in the even and odd case. This is Lennart's comment on the Hairy Ball theorem.
I think that this tactic can be made to work for all manifolds that admit metrics all of whose geodesics are closed (and of the same length).
If one feels like it one can even compute string operations like the Chas-Sullivan product in this manner.
If my memory serves me right, this is neatly discussed by W. Ziller when he dicusses the geometry of the Katok examples. In the case of the two-sphere, Katok constructed an easy example of a Finsler metric that has only two prime geodesics. Thus the Morse theory of its energy functional on the loop space is particularly simple and can be used to understand its topology.
