ESNs, MEIDs, and IMEIs are all codes used by manufacturers and cellular carriers to uniquely identify each device. Each device will generally have one MEID, ESN, or IMEI.

Your device identifier can be found via Android's settings: Settings > About Phone > Status > IMEI information (or similar)

More information can be found through the Wikipedia links below:

Your device identifier (i.e., MEID, ESN, IMEI) can be used to check with a device's carrier (e.g., Verizon, T-Mobile, Sprint, AT&T) to check if the device is "clean" (not reported lost or stolen) and not tied to an account in bad standing.

Related Questions