These are dead-tree printing terms used for offset printing. Offset printing is where the inked image is transferred (=offset) from plate to rubber sheet to paper.
Process colour = creating a colour through layers of four specific inks (CMYK). CMYK is cyan (blue), magenta (red), yellow and key (black).
Spot colour = a specially pre-mixed ink, can include metallic and fluorescent colours. The biggest spot colour system is Pantone. Spot colours are used when it is important for a colour to be consistent throughout a document, or when accuracy is important (like in a logo).
When printing people talk about process and spot colours (so a process plus two spot colour document would be a six colour job). In fact, in offset printing one single run with one ink on the plate and rubber sheet will be one spot colour – so CMYK are actually also four spot colours. This is just to confuse you though: In the real printing world colours are defined as either spot or process.
This About.com article is really helpful:
When to Use Spot Colors or Process Colors or Both