A manifold is usually by default assumed to have an underlying topological space which is a Hausdorff topological space. If that condition is explicitly dropped, one accordingly speaks of a non-Hausdorff manifold. Compare the red herring principle.
Assuming that we're talking about topological manifolds, a non-Hausdorff manifold is still T-1? and sober.
The usual example is the real line with the point $0$ ‘doubled’. Explicitly, this is a quotient space of $\mathbb{R} \times \{a,b\}$ (given the product topology and with $\{a,b\}$ given the discrete topology) by the equivalence relation generated by identifying $(x,a)$ with $(y,b)$ iff $x = y \ne 0$.
Mathieu Baillif, Alexandre Gabard, Manifolds: Hausdorffness versus homogeneity (arXiv:0609098)
Steven L. Kent, Roy A. Mimna, and Jamal K. Tartir, A Note on Topological Properties of Non-Hausdorff Manifolds, (web)