Could someone explain how to construct a product lattice, or point me to an explanation on the web?

As I explain in this MO answer, there is a choice of product structures to place on the product of two lattices. Please click through and read the discussion there. But to summarize, one choice is to form the product order, where you consider pairs $(a,b)$ and the orders are inherited in each coordinate separately. Another choice is to use the lexical order on pairs, but this only results in a lattice when there are certain completeness or linearity assumptions. The lexical product $K\times L$, however, amounts to replacing each node in $K$ with a copy of $L$, and this is often what is desired. See also this question for a discussion of the completeness issue for the lexical order. 


Look at http://en.wikipedia.org/wiki/Lattice_%28order%29#Examples under the example it calls the Cartesian square of the natural numbers. This tells you enough about what the order will be, and the meet and join are the obvious things (meets and joins of components). 

