What are the properties that hold for a fundamental group of a surface and does not hold necessary for the fundamental groups of manifolds of higer dimensions ??

This question is very vague, but here are some thoughts to add to Mark's answer. First, note that any finitely presented group arises as the fundamental group of a closed manifold of dimension 4 (see this MO question), which is a huge contrast to the very special case of dimension 2. The properties of the fundamental groups of 3manifolds are a subject of very active research, much aided by Perelman's solution to the Geometrisation Conjecture. Like the 2dimensional case, 3manifold groups are residually finite (a theorem of Hempel). The fact that there is no closed 3manifold with every infiniteindex subgroup free is only very recent known, as a result of work of Kahn and Markovic. I don't think any closed 3manifold has cohomological dimension 2, so that property actually does it on its own. 


It's a conjecture that surface groups are characterized by being the only 1relator groups such that every finiteindex subgroup is also 1relator and every infinite index subgroup is free. 


Every subgroup of infinite index is free, the group is residually finite, and the cohomological dimension is 2. 


The word problem for the fundamental group of a closed surface is solvable, using Dehn's algorithm. Since any finitely presented group appears as the fundamental group of some closed $4$manifold, and there are such groups for which the word problem is unsolvable, this is indeed a special property for two dimensions. 


A surface group is either virtually abelian, or word hyperbolic (or both, when it is finite). In some sense, this reflects the fact that every surface admits a Riemannian metric of constant curvature, and that the sign of the curvature is detected by the fundamental group. In dimension 3, Perelman's uniformization implies that compact manifolds can be decomposed into "geometric pieces" (that are again detected in a suitable sense by their fundamental groups), while in higher dimension there is no hope for a simple result of this type. 

