I can't resist.
Urysohn (followed by Bing and others) has constructed a connected Hausdorff space which has only countably many points but more than one. Of course, such spaces are separable.
On the other hand, every connected component of an arbitrary countable metric space has only one point.
Also, countable connected spaces cannot be Hausdorff compact since every connected component of an arbitrary Hausdorff compact countable space has only one point.