Would it be worth it to mine with a cheap laptop?
Generally Laptop mining is not worth the stress (heat) that the video card. Even some of the best laptop gpus ($1500+) only get 20 - 40 mhash compared to a $300 cheapo computer with a 5770 with 200 mhash.
How exactly does it all work?
in short:
Install the latest ATI driver package 11.7 or above
install the SDK 2.1 or 2.4 with opgenCL
download a bitcoin miner (for windows guimminer is easiest)
decide whether you want to solo mine or pool mine. (Pooled mining is prob the best solution here)
create an account with a pool miner operator. (BTC guild, slush's pool, deepbit...)
start mining
How much electricity would it take up?
depends on the power supply and video card solution
And is it worth it?
It depends. if your goal is to make money then you need a desktop platform.
here is list of hardware
https://en.bitcoin.it/wiki/Mining_hardware_comparisonhere is a list of builds
https://en.bitcoin.it/wiki/Mining_rig