A manifold is a topological space which locally looks like R^n. Calculus on a manifold is assured by the existence of smooth coordinate system. A manifold may carry a further structure if it is endowed with a metric tensor. Why further structure? If have sphere or a cylinder I can parameterize it with different coordinates ( two patches for sphere), which means different metrics. But does not have any manifold a canonical metric from the start? What is meant with endowing a manifold with a metric? If I got a cylinder, its metric is euclidean by definition, no need for defining a extra structure on it.