A block is a group of Magic: The Gathering expansion sets that are centered or focused on a common storyline cycle, or arc. Typically, from Mirage onwards the first set in a block was a large expansion, and was followed by two small expansions; but, this trend was subverted a number of times (e.g., with Lorwyn block and Shadowmoor block, which comprised two expansions each; Return to Ravnica block, which comprises two large expansions and one small expansion).   
Beginning in the Fall of 2015, blocks will be two sets each. Each block will have one large set that introduces the world and one small set that builds upon it (this is the default and WotC reserves the right to have sets of different sizes if it serves the block). Blocks will no longer have a third set. There will be a large fall expansion and small winter expansion set in World #1. Then there will be a large spring expansion and a small summer expansion, usually set in World #2. The first two sets will be part of one block and the second two sets will usually be part of a second. 
Modern block design commenced in 1997, with Mirage block.
Ice Age and Alliances were grouped together with Homelands as a block retroactively. As Homelands is thematically and mechanically unrelated to Ice Age and Alliances, as well as being of a different storyline arc, it was later removed from the Ice Age block and replaced with Coldsnap in 2006. 
One of the most important parts to designing and developing sets today is to create cross-block synergies so the sets within Standard play well with each other, but also so that there is enough of a change when Standard rotates to change things up. A diverse and shifting metagame is a healthy metagame. 
The redesign of the block structure from three to two sets (the so-called Two Block-Paradigm; i.e. two blocks each year) was announced by Mark Rosewater on August 25, 2014, and will take effect in September 2015 with the release of "Blood". 
List of blocks
The following list details all of the blocks in Magic: The Gathering in chronological order.
The year given in parentheses is the year in which the first set in the block was released. Typically, with a notable exception being Lorwyn and Shadowmoor blocks, Wizards of the Coast has released the first set in October of the year in parentheses and subsequent sets in the block in February and April of the following year.
- Mark Rosewater. (December 07, 2009.) "Playing With Blocks", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Mark Rosewater. (April 29, 2013.) "Third Time's a Charm", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Sam Stoddard. (May 17, 2013.) "Building Blocks", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Mark Rosewater. (August 25, 2014.) "Metamorphosis", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Blake Rasmussen. (August 25, 2014.) "Building Blocks", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Sam Stoddard. (October 11, 2013.) "Cross-Block Synergies in Theros", Daily MTG, magicthegathering.com, Wizards of the Coast.