IMSI = International Mobile Subscriber Identity. This is a unique identifier that defines a subscriber in the wireless world, including the country and mobile network to which the subscriber belongs. It has the format MCC-MNC-MSIN. MCC = Mobile Country Code (e.g. 310 for USA); MNC = Mobile Network Code (e.g. 410 for AT&T), MSIN = sequential serial number. All signaling and messaging in GSM and UMTS networks uses the IMSI as the primary identifier of a subscriber.
The IMSI is one of the pieces of information stored on a SIM card.
ICCID = Integrated Circuit Card ID. This is the identifier of the actual SIM card itself - i.e. an identifier for the SIM chip. It is possible to change the information contained on a SIM (including the IMSI), but the identify of the SIM itself remains the same.
IMEI is short for International Mobile Equipment Identity and is a unique number given to every single mobile phone, typically found behind the battery.
IMEI numbers of cellular phones connected to a GSM network are stored in a database (EIR - Equipment Identity Register) containing all valid mobile phone equipment.
When a phone is reported stolen or is not type approved, the number is marked invalid.
MSISDN = Mobile Station ISDN number. This is the full phone number of a subscriber, including the national country code (e.g. 1 for US, 44 for UK, etc.). The purpose of the MSISDN is simply to allow a device to be called. A subscriber can have multiple MSISDNs (e.g. one phone number for business, one for personal calls, one for fax, etc.), but generally only one IMSI. The MSISDN does not need to be stored on the SIM card. In cases where it is stored on the SIM, the main reason is so that the user can use check to see what their own MSISDN is (in case they forget). The MSISDN is never signaled to of from the device.