There are several ways to describe the coordinate system using gdalwarp. In the example, it uses the proj.4 nomenclature, but you could also use the EPSG code or the name of a .prj file.
the proj.4 code in the example includes information about the projection type (utm= Universal Transverse Mercator projection), the projection parameter (with UTM, giving the zone number - 11 - is sufficient to describe the projection, but usually you need more parameters) and the geographic coordinate system (datum) used as a base for the projection (WGS84 is a standard GCS, so it is sufficient to provide its name alone).
In practice, I usually copy-paste the string from a spatialreference.org and modify some of the parameters from a similar projection if needed. Most of the world's projection are available on this site (typing EPSG + name of the projection is another way to quickly find the string that you need).
Note that it is recommended to specify both source and target references for accurate results.