You are asking about one of the most fundamental questions in topology, still very far from being resolved in full.
For 2-manifolds see here for a complete classification.
Then it gets complicated.
For 3-manifolds see here for an overview. The question of whether there is a "complete" classification might be regarded as answered, but it might also be regarded as still open since no-one has really quite described the complete list yet.
Then it gets impossible.
For manifolds of dimension $\ge 4$, there is a lot of deep theory, and many partial results, but a complete classification seems still far beyond our present capabilities. See here for an overview of 4-manifolds, which has many special features. See here for a very brief description of manifolds of dimension $\ge 5$, which has several kinds of coherent theories independent of dimension, although the actual answers provided by those theories do depend on dimension.