What is a surface? Let's cut or cleave a bulk crystal and see what could possibly happen:

If the surface is a metal, the free electrons in the top layer are not confined to their unit cell by a neighboring cell any more and will spill out into the vacuum. As a result the bond is weakened and the ion cores will move a bit outward, too. This effect is called relaxation.

The situation at semiconductor surface is quite different: here we have strongly directional bonding and the broken bonds at the surface layer, may handshake with a neighboring dangling bond. As a result, for instance pairs of surface atoms are formed, and the lattice period doubles. Such a change of the in-plane structure is called reconstruction.

Sometime nothing happens as we cleave the crystal. This is typical for layered materials like graphite or TaSe2. Also many small molecules organize in layers that are only weakly bound to each other.

The details of what happens depend on the specific material. For instance many transition metal surfaces show some kind of reconstruction. These structural changes at the surfaces affect the elctronic states at the surface and hence the chemical properties. Hence for an understanding of catalysis or transistors surface states are of great importance.
 

There are 5 two-dimensional Bravais lattices:

Often it is useful to choose a different unit cell for the surface. As an example we show the usual choice of surface unit cells for the densely packed fcc surfaces:

The next question to ask is, what happens when other atoms react with a surface. A great wealth of so-called superstructures have been found at surfaces. The larger unit cells arise because of special constaints the adsorbed atoms face due to bonding, or simply, because the adsorbed atoms are too large or too small to follow the substrate period. As examples we show the simplest kinds of superstructures on a square lattice:

Determining the surface unit cell is the first step in the understanding of the surface. In order to determine the surface structure the adsorbate site has to be identified. For a square surface the high-symmetry sites are the on-top site, the fourfold-hollow site, and the bridge site, however, other sites have been found, too.