For a short and painless description of how ultraproducts are colimits see page 6 of this article (it is equivalent, but I found it more readable than the descriptions given in the answers):

H. Mariano, F. Miraglia, _Profinite structures are retracts of ultraproducts of finite structures_, Reports on Mathematical Logic **42** (2007) pp169-182 ([abstract](http://rml.tcs.uj.edu.pl/rml-42/a-mar-42.htm), [pdf][1])

The article contains a nice application of this categorical description of ultraproducts...

  [1]: http://www.asc.tuwien.ac.at/~herfort/W_Hojka/Mariano_et_al_profinite_structures.pdf