Classification of manifolds

"Low dimensions" means dimensions up to 4; "high dimensions" means 5 or more dimensions. The case of dimension 4 is somehow a boundary case, as it manifests "low dimensional" behaviour smoothly (but not topologically); see discussion of "low" versus "high" dimension.
More precisely, what is the structure of the set of additional structures?

In more general categories, this structure set has more structure: in Diff it is simply a set, but in Top it is a group, and functorially so.

There are two usual ways to give a classification: explicitly, by an enumeration, or implicitly, in terms of invariants.

The point-set classification is basic—one generally fixes point-set assumptions and then studies that class of manifold. The most frequently classified class of manifolds is closed, connected manifolds.

Being homogeneous (away from any boundary), manifolds have no local point-set invariants, other than their dimension and boundary versus interior, and the most used global point-set properties are compactness and connectedness. Conventional names for combinations of these are:

Similarly for 3-manifolds: of the 8 geometries, all but hyperbolic are quite constrained.

One can take a low-dimensional point of view on high-dimensional manifolds and ask "Which high-dimensional manifolds are geometrizable?", for various notions of geometrizable (cut into geometrizable pieces as in 3 dimensions, into symplectic manifolds, and so forth). In dimension 4 and above not all manifolds are geometrizable, but they are an interesting class.

Conversely, one can take a high-dimensional point of view on low-dimensional manifolds and ask "What does surgery predict for low-dimensional manifolds?", meaning "If surgery worked in low dimensions, what would low-dimensional manifolds look like?" One can then compare the actual theory of low-dimensional manifolds to the low-dimensional analog of high-dimensional manifolds, and see if low-dimensional manifolds behave "as you would expect": in what ways do they behave like high-dimensional manifolds (but for different reasons, or via different proofs) and in what ways are they unusual?

There is a unique connected 0-dimensional manifold, namely the point, and disconnected 0-dimensional manifolds are just discrete sets, classified by cardinality. They have no geometry, and their study is combinatorics.

Four-dimensional manifolds are the most unusual: they are not geometrizable (as in lower dimensions), and surgery works topologically, but not differentiably.

Since topologically, 4-manifolds are classified by surgery, the differentiable classification question is phrased in terms of "differentiable structures": "which (topological) 4-manifolds admit a differentiable structure, and on those that do, how many differentiable structures are there?"

In dimension 5 and above (and 4 dimensions topologically), manifolds are classified by surgery theory.

For maps, the appropriate notion of "low dimension" is for some purposes "self maps of low-dimensional manifolds", and for other purposes "low codimension".

Particularly topologically interesting classes of maps include embeddings, immersions, and submersions.