AT&T is GSM, and so their phones use SIM cards. Verizon and Sprint are CDMA, and so theirs do not. Switching phones with a SIM is as simple as removing the card from one phone and putting it into another. With CDMA, you need to contact the carrier to switch phones.
A mini SIM card (used by the iPhone 4 and 4s) is essentially a cut down version of a regular SIM. They're not common yet (but, presumably, they will replace regular SIM cards over time).
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AT&T is GSM, and so their phones use SIM cards. Verizon and Sprint are CDMA, and so theirs do not. Switching phones with a SIM is as simple as removing the card from one phone and putting it into another. With CDMA, you need to contact the carrier to switch phones.
A mini SIM card (used by the iPhone 4 and 4s) is essentially a cut down version of a regular SIM. They're not common yet (but, presumably, they will replace regular SIM cards over time).