At present there are three main formats in the World of Warcraft trading card game that are widely played: Classic, Core, and Block.
Classic Constructed is a best-of-three format, played over the course of one hour. All cards that have been released are legal for play in a Classic Constructed deck; there is no ban list.
Core Constructed is a best-of-one format, played over the course of half an hour. Only cards from the last three blocks (detailed below) are legal for play in a Core Constructed deck; there is no ban list.
With Core Constructed, individual sets do not become illegal for play each time a new set is released; when a new block is started, the oldest entire block will rotate out of Core legality.
On the 14th December 2010 Worldbreaker - the first set of Block 5 - will be released, and with that, the entirety of Block 2 will no longer be legal for Core Constructed play.
Block Constructed is usually played once all three sets of a block have been released. It is played best of three over an hour, and only cards from the specified block are legal for play. You can choose to play Block Constructed with any complete block, though for competitive play, usually the newest block is used.
Here's a breakdown of what sets, raids and boutique releases are in which block.
Heroes of Azeroth [AZEROTH]
Through the Dark Portal [DARK PORTAL]
Fires of Outland [OUTLAND]
Feast of Winterveil set cards (from Feast boxed set) [WINTERVEIL]
Onyxia's Lair treasure [ONYXIA]
Molten Core treasure [MOLTEN CORE]
The Burning Crusade promo cards (from WoW Burning Crusade Limited Edition) [BC PROMO]
Mrglrglmrglmrrrlggg [BLIZZCON 2007]
March of the Legion [LEGION]
Servants of the Betrayer [plus Betrayer Crafted, BETRAYER]
Hunt for Illidan (plus Illidan Crafted) [ILLIDAN]
Magtheridon's Lair treasure [MAGTHERIDON]
Black Temple treasure and Traitor cards [BLACK TEMPLE]
The Darkmoon Faire [DARKMOON FAIRE]
Crafted 1-14 (various icons) [CRAFTED]
Drums of War (plus Drums Crafted) [DRUMS] [DRUMS CRAFTED]
Blood of Gladiators (plus Gladiators Crafted) [GLADIATORS] [GLADIATORS CRAFTED]
Fields of Honor (plus Fields Crafted) [HONOR] [HONOR CRAFTED]
Drums of War Starter [DRUMS STARTER]
Death Knight Starter [KNIGHT]
Arena Grand Melee [GRAND MELEE]
Badge of Justice 1-4 (little diamond-y icon) [BADGE]
Scourgewar (plus Scourgewar Badge 1-4 and Scourgewar Crafted) [SCOURGEWAR] [scourgewar icon, CRAFTED] [scourgewar icon, BADGE]
Wrathgate (plus Wrathgate Badge 1-4 and Wrathgate Crafted) [WRATHGATE] [wrathgate icon, CRAFTED] [wrathgate icon, BADGE]
Icecrown (plus Icecrown Badge 1-4 and Icecrown Crafted) [ICECROWN] [icecrown icon, CRAFTED] [icecrown icon, BADGE]
Class Starter Decks [CLASS]
Naxxramas treasure [NAXXRAMAS]
Feast of Winterveil Promo Card (Red Rider Air Rifle) [HOLIDAY 1]
Class starters (Block 4) are a bit of a strange one - they contain cards from a variety of older sets, but they are classed as 'reprints' and have new set numbers [CLASS xx/234] compared to the original cards which had their original set number on.
Any card that has been reprinted is considered part of Block 4, and either the original card or the card from the Class Starters can be used - all cards are functionally identical, they just carry a different number along the bottom of the card.
For a list of which cards were reprinted in the Class Starters, take a look at this post on the official WoWTCG forum, by Simon Key.
Archives was released in late 2010, and contains foil versions of ~150-200 cards from throughout the game's history. These cards carry their original set numbers, and are not classed as reprints. Any card that is not Core Constructed legal in its original form is also not Core Constructed legal in its foiled form.