AT&T uses the 850Mhz and 1900 Mhz bands for GSM (voice), EDGE (2G) and UMTS (3G) data. In some markets they use both bands in others one or the other. Other than some outdated maps, there's no way to tell where they use each band.
For LTE AT&T uses bands 2 (1900), 4 (1700), 5 (850), 12/17 (700 a, b) 29 (700 d) and 30 (2300) Band 12 is AT&T's primary LTE band. Band 17 is the subset of band 12 that AT&T uses. AT&T originally identified its 700 Mhz spectrum as band 17 but now identifies it as both 12 and 17 which allows phones with either band 12 or band 17 to use it. Bands 2, 4, 5, 29 and 30 are used in areas where AT&T doesn't own any band 12 spectrum and also in combination with band 12 in urban areas to increase capacity and speeds.
T-Mobile uses the 1900 Mhz band for GSM (voice) and EDGE (2G) and UMTS (3G) data. It also uses the 1700 (AWS) band for UMTS (3G) data in some markets.
For LTE, T-Mobile uses bands 2 (1900), 4 (1700) and 12 (700 a,b,c). T-Mobile's primary LTE band is band 4. Band 2 is being deployed in former 2G only non-urban areas where T-Mobile doesn't have any band 4 spectrum. In some, maybe all, of these band 2 only LTE markets, 3G is not offered, only 2G and LTE.
Band 12 is relatively new. T-Mobile started deploying LTE on band 12 in the second half of 2014. There's a map of T-Mobile's band 12 spectrum here. Band 12 is a lower frequency band that provides extended range and improved building penetration compared with bands 4 and 2.
As for Sprint and Verizon, you can pretty much forget about using phones from other operators on either of their networks. With a couple of exceptions (iPhones from the Apple Store and the Nexus 5 from the Play Store), both operators will only activate phones that they have sold. It is still possible to "flash" Sprint, U.S. Cellular and other non-Verizon CDMA phones to Verizon MVNO PagePlus, but even that avenue is expected to be blocked soon. It's still useful to know which bands Sprint and Verizon use when shopping for phones as older phones nay not support all of the bands currently used by either operator.
Verizon uses 850 Mhz and 1900 Mhz for voice, 1xRTT (2G) and EVDO (3G) data
For LTE, Verizon uses bands 4 (1700c) and 13 (700). Band 13 is Verizon's primary LTE band. Band 4, which Verizon markets as XLTE, is used to provide increased speed and capacity in urban areas.
Sprint uses the 800 Mhz and 1900 Mhz bands for voice, 1xRTT (2G) and EVDO (3G) data
For LTE Sprint uses bands 25 (1900), 26 (800) and 41 (2500). Band 25 is Sprint's primary LTE band. Bands 26 and 41 are used to increase capacity and speed. Sprint markets service using all three LTE bands as Sprint Spark.
WiMAX, Sprint's original 4G technology, operates in only 71 of the over 300 US mobile markets. It uses the 2.5Ghz band and is scheduled to be shutdown at the end of 2015.
To be sure that a phone will work to its fullest or at all with a given carrier you need to know which bands it supports.
For phones made for the US market PhoneScoop.com is my favorite resource for checking specs. It provides a full page of detailed specifications for virtually every US phone model. Supported bands are listed in the Modes section of the specs. Click the + more detail link to see which LTE band numbers are supported. With LTE It's not enough to refer to bands by their approximate frequency (850, 1700, 1900), as multiple incompatible LTE bands can exist in the same frequency ranges. LTE bands can only be positively identified by their LTE band number.
For checking band support for non-US market phones I like PDAdb.net and frequencycheck.com Be sure you are looking at specs for the phone's exact model number rather than its marketing name (Samsung Galaxy i9500 rather than Samsung Galaxy S4), as most phones come in multiple versions supporting different bands.
When comparing phones you will quickly discover there are relatively few truly universal phones that are sold unlocked and work across all operators. The only ones I'm aware of:
Apple iPhone 6s (A1633) and 6s Plus (A1634)
Nexus 6P (H1511)
Samsung Galaxy S7 Special Edition (SM-G930U) and S7 Edge Special Edition (SM-G935U)
Verizon iPhone 6 (A1586) and 6 Plus (A1522) lack T-Mobile band 12 and AT&T band 30
iPhone SE (A1622) lacks Sprint band 41
Nexus 5x (H790) and Nexus 6 (XT1103) lack AT&T band 30
Motorola Moto X Pure Edition (2105) (XT1575), Moto G4 (SM-G935U) Moto G4 Plus (XT1644) and Moto G4 Play lack AT&T bands 29 and 30
Updated 27-Aug-2016 - revised AT&T and T-Mobile band lists and descriptions, updated "universal" phone list.
Sources: Cellular Binder, TmoNews, Apple, Wikipedia
Image: Cell phone tower in Oregon by M.O. Stevens some rights reserved CC-BY-3.0 via Wikimedia Commons