I've read about a "pancake lens" but don't really understand what that means.
What the pros and cons of having a pancake vs. another type of lens?
I'm thinking of getting a Panasonic Lumix DMC-GF1 which comes with such a lens.
A pancake lens is designed to be physically very thin and compact, being relatively pocketable when combined with a slim body.
Current pancake lenses have a fixed focal length in the middle of the spectrum (neither wide nor telephoto), and they are relatively slow for a prime lens (often f/2.8) although exceptions exist. The image from a pancake lens looks the same as an image from a bulkier lens of the same focal length; they have no special effect.
Although the pancake design is quite old, it has been popularized lately by mirrorless systems. Currently produced pancake lenses include:
Here is an example of a pancake lens:
Canon EF 40mm f/2.8 STM pancake lens. Wikimedia Commons, CC-BY-SA
"Pancake lens" is a purely physical description.
If the lens is significantly wider than it is long, it's a pancake lens.
Actual overall size has nothing to do with it, and it has no implications about image performance, aside from the fact that it's generally very hard to pack a telephoto or fast prime into the pancake form-factor.
It's worth noting that a "Pancake lens" does not have to be small or compact, and a small lens is not automatically a pancake lens. It's only a description of the lens form-factor.