What is a topological manifold and why do we need coordinate charts?

I'm starting to learn about manifolds. The definition is:

A topological space $M$ is an $n$-dimensional topological manifold if:

1. $M$ is Hausdorff
2. $M$ is second-countable
3. $M$ is locally Euclidean: each point has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$

The homeomorphisms $\varphi: U \to \varphi(U) \subseteq \mathbb{R}^n$ are called coordinate charts.

Why do we need the Hausdorff and second-countable conditions? Can someone give an example of a locally Euclidean space that is NOT a manifold because it fails these conditions? And what role do charts play in defining calculus on manifolds?