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