What is the analogue of a covering map in algebraic geometry?

Question

It is well known that many of the concepts of algebraic geometry have some corresponding concept in differential manifolds. Take for example, the idea of a tangent space, or of cohomology of smooth manifolds/schemes. Specifically, many properties of morphisms in algebraic geometry have direct analogues in differential topology -- such as proper morphisms/maps, or embedding of manifolds/immersion of schemes.

So is there any type of morphism of schemes, or of varieties, that can be intuitively considered analogous to a covering map? Furthermore, is there any sort of concept of a "universal cover" of a variety, some cover that is particularly well behaved (simply connected, in the case of smooth manifolds)?

Answer 1

The equivalent of covering in algebraic geometry is the notion etale cover. There is not a notion of universal cover, but the etale cover of $X$ define the etale fundamental is a projective limit of finite groups.

Answer 2

is there any sort of concept of a "universal cover" of a variety, some cover that is particularly well behaved

In addition to the coverings associated with the SGA fundamental groups there is the Vakil-Wickelgren universal cover, http://arxiv.org/abs/0902.3464 and the recent work of Scholze and Bhatt on the pro-etale topology.