What are topological manifolds for you?

In this question different people understood different things when talking about topological manifolds. Some argued they they have to be Hausdorff, some that they have to be second countable and some, both.

When I studied them, my teacher showed us examples of non-Hausdorff (the line with two origins) and non-second countable (the long line) manifolds. For me, a topological manifold is a locally Euclidean topological space.

What are the different definitions of a topological manifold you know? What it depends on? What author you read when you studied them? Who was your teacher?

EDIT: What properties have the topological manifolds if we define them as second-countable and Hausdorff that they don't have if they are only locally Euclidean?

Hausdorffness is a necessary condition for continuous real-valued functions to separate points. If continuous real-valued functions don't separate points on your space, you're dealing with a weird space (which in particular does not embed into $\mathbb{R}^n$). Of course weird spaces exist, but it's harder to prove nice theorems about them.

As Mark Schwarzmann says in the comments, any manifold that embeds into $\mathbb{R}^n$ is necessarily both Hausdorff and second-countable.

It is also common to see paracompactness as a condition, since this is equivalent (in the presence of the other axioms) to both metrizability and the existence of partitions of unity, which I understand to be quite useful.