Sometimes, the Illumina website descriptions aren't too bad[1-2]. Unfortunately, I don't know of any great resources targeted to a lay audience (which is not to say they do or don't exist). I work in a lab that does a lot of high throughput sequencing, so when I have questions about particular assays, usually someone in lab knows, and if not someone else in the building does.

Mate-pairs are essentially reads taken from opposite ends of a single fragment of DNA. This is useful for de novo sequencing, where they act almost like a single, very long read. The problem with random sequencing (both shotgun sequencing and current "next-generation" sequencing technologies) is that it's not obvious how a given read relates to the other reads you have.

Using the jigsaw puzzle analogy, mate-pair sequencing is roughly equivalent to being told that a given piece is 8-10 pieces away from another piece in a given direction. You still won't necessarily know exactly where to put either of the pieces, but it narrows the search space hugely, especially when you have one in place.

[1] Mate Pair libraries: http://www.illumina.com/technology/mate_pair_sequencing_assa...

[2] Paired-end sequencing: http://www.illumina.com/technology/paired_end_sequencing_ass...