There are many dimensionality reduction techniques, the most famous being Principal Component Analysis which in fact works quite well when the data points are somewhat linearly distributed. However it produces disastrous results in case of non-linear distributions.
What is a manifold? The underlying idea is that in very large dimensional spaces, data points of physical processes usually tend to occur in restricted regions, and are almost never uniformly distributed through the entire space. A manifold is an imaginary surface on which these data-points are assumed to lie. Furthermore the key concept is that this manifold has very few degrees of freedom, and hence can be represented in a low-dimensional space.
The distance between the data points is now computed as the shortest distance while tracing along the manifold. For example, consider our Earth. To go from Delhi to New York you could drill through the ground and obtain the shortest distance in 3-D Euclidean space. However this is infeasible and the true shortest distance is actually the geodesic distance, the one when traveling on the surface of Earth. Here our Earth’s surface is our manifold in the large universe!
Given the data points, the objective is to find this manifold so that data-points close to each other on the manifold surface can be classified as belonging together instead of the ones which are close in Euclidean space. However finding this exact geometric shape is truly a Herculean task in most cases. Hence, the objective is transformed to find a function which can map the data from the high dimensional space to a simpler low dimensional space, such that the distance between the points in the low dimension corresponds to the geodesic distance as computed on the manifold.