A topological manifold is just a topological space with certain required properties (depending on the author / context this could be Hausdorff and second countable for example) and the ubiquitous property of being locally euclidean i.e. every point has a neighborhood with a homeomorphism taking that neighborhood to an open subset of euclidean space; this pair is called a coordinate chart and the set of all coordinate charts on a topological manifold is called an atlas. Because of the locally euclidean property, amongst other things e.g. second countability, manifolds possesses many nice properties such as being locally path connected (which for manifolds is equivalent to being locally connected), sigma compact, and locally compact to name a few.
We can go can an extra mile and demand that the atlas be smooth in the sense that if two coordinate domains happen to over lap, the transition map taking the image of one coordinate domain to the image of the other is smooth in the usual sense of calculus. One usually requires that these smooth atlases be maximal. This then allows us to make unambiguous sense of smoothness of maps by looking at the local coordinate representations on coordinate charts and just talking about calculus in the usual sense in euclidean space. Of particular importance is that now we can make sense of a differentiable curve on a manifold and this is one way to then talk about tangent vectors. We can also go on to define vector fields, covector fields, and tensor fields all of which are, for example, crucially needed objects in physics.
There is another structure we can define on smooth manifolds and this is called an affine connection. Loosely put, it allows us to "relate" vectors at different points on a manifold because, unlike in euclidean space, one cannot simply parallel translate vectors to the same tangent space in a trivial way. Using affine connections, we can talk about a kind of directional derivative along the tangent vector at a point on a curve, called a covariant derivative; using this we can then make sense of what it means to parallel transport a vector along a curve from one tangent space to another. We can also then define an affine geodesics as a curve where the covariant derivative of the tangent vector along itself is zero at every point on the curve.
The final thing we would like to introduce is a riemannian metric. This is a map that assigns to each tangent space at every point an inner product. It is NOT, and I want to really stress NOT, the same thing as the metric from analysis which allows you to measure distances between points in a metric space. A riemannian metric gives you an inner product for each tangent space. Associated with every riemannian metric is a unique, torsion free (meaning covariant derivatives commute on scalar valued functions) connection called a levi - civita connection. Loosely put, the levi civita connection preserves the inner product of two vectors as you parallel transport them along some curve from one tangent space to another. We can use the riemannian metric (also called metric tensor) to define lengths of geodesics and the levi - civita connection to define the riemann curvature tensor (there are other kinds of curvatures we can employ such as Ricci curvature, the scalar curvature associated with the Ricci curvature, sectional curvature etc.).
Putting all these together gives you the framework that Einstein's theory of General Relativity is built on (which was a major, major application of riemannian geometry back in the day) wherein a space - time is a smooth manifold equipped with a pseudo - riemannian (we don't require positive definite) metric. Local experiments done by an observer should be able to recover the laws of special relativity and locally observers cannot detect gravity which manifests itself as the curvature of the manifold. Test particles in free fall (free from all non gravitational interactions) follow geodesics. Manifolds are also very important in classical mechanics where one describes the configuration space of a system by a smooth manifold with a symplectic 2 - form (that is related to the Hamiltonian of the system; the non degenaracy allows us to write down and solve Newton's 2nd law in terms of this Hamiltonian) and the cotangent bundle becomes the phase space. Anyways, I hope you enjoy the subject, it is very very cool! I only know the subject from a physics standpoint but it is all very elegant nonetheless. I'll stop rambling now =D.
