An ESN is an Electronic Serial Number. On Analogue, TDMA, and CDMA phones, this is either an 11 digit number, or (especially with Motorola handsets) an 8 digit hex expression.
The ESN should not be confused with the IMEI, which is the serial number for GSM phones, and which is 11 digits, and does not serve the same purpose at all.
TDMA and CDMA phones are mostly used in the Western Hemisphere, since folks in the East quite rightly decided early on that GSM was a better option.
The ESN serves a number of purposes. First, it helps to identify what kind of phone you have, if it's not immediately obvious. It can be found on the back of the phone underneath the battery. Most cell phone carriers will have a database of phones they've sold, from which they can look up the ESN and tell you what kind of phone you have, as well as how to program it, or whether to use it as a paperweight and upgrade.
Additionally, every time you try to make a call with a TDMA or CDMA phone, the system checks your ESN and a number called an authentication key to decide whether your number actually is who it says it is, and should be allowed to call out on that network. If the network loses track of your ESN, you will become unauthenticated and will not be able to call out.
Most CDMA and TDMA programming lists the ESN first on the programming screen, but this is just another way of viewing it if you can't get the back off of the phone...you can't change it.