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?