The Delaunay triangulation is a natural and popular way to triangulate the convex hull of a set of points, $P$, in the plane.
Conceptually, we can construct it as the dual of the Voronoi diagram of $P$. The Voronoi diagram partitions the plane into Voronoi cells, one for each point in $P$. The Voronoi cell for a point $p \in P$ is defined as that portion of the plane consisting of points that are at least as close to $p$ as to any other point in $P$. The Voronoi cells are convex polygons, but points on the convex hull of $P$ have unbounded Voronoi cells.
If two points are Voronoi neighbours, i.e., their Voronoi cells share a boundary edge, then they will be connected by an edge in the Delaunay tessellation. A Voronoi vertex is a vertex of the polygon describing a Voronoi cell. A Voronoi vertex belongs to at least three Voronoi cells. If there is no Voronoi vertex that is shared by more than three Voronoi cells, then the Delaunay tessellation will be the unique Delaunay triangulation (in which case we say the points are in general position). Otherwise, the Delaunay tessellation will contain higher order polygons, which may be triangulated arbitrarily to obtain a Delaunay triangulation.
The Delaunay triangles have the property that the inside of their circumcircles contain no points from $P$ (this is easy to check from the above description). Also, of all possible triangulations of $P$, the Delaunay triangulation maximizes the minimum angle that can be found in a triangle (this does not follow easily from the above description -- it is usually demonstrated via an edge flipping argument).
To construct the Delaunay triangulation in practice, there are several efficient algorithms. There are also many freely available implementations of these algorithms. CGAL is one popular example.