The "legend rule" is the name for a rule that prevents a single player from having two or more of the same legendary permanent in play at the same time. This latest version of the rule has been in effect since the release of Magic 2014.  
If the same player controls two or more legendary permanents of the same name, that player will choose one of those permanents, and the others are sent to their owners' graveyards when state-based effects are checked. This is not a destruction effect, and can't be prevented by regeneration or indestructibility.
Currently, only two cards circumvent the "legend rule." Brothers Yamazaki allows up to two of them to be in play at the same time (though a third will send at least one to the graveyard, depending on how they are controlled), and Mirror Gallery cancels out this rule.
Explanation of the latest change
One way to think about the new "legend rule" is that it now looks at each player individually. It doesn't matter what any other player controls. Clearly, this has a few play ramifications. If you control a legendary permanent, having another one enter the battlefield (by playing a second one or creating a copy of the first one) will leave you with one on the battlefield. It may be the old one. It may be the new one. That's up to you.
Also, creating a copy of a legendary permanent controlled by another player will simply give you a copy. The one controlled by the other player won't explode, won't leave the battlefield, and really won't be affected at all. Clones do what they were intended to do, which isn't to be situational killing machines.
Previous versions of the "Legend Rule"
From Legends to Champions of Kamigawa
Besides the connection to the subtype (later to be changed to a supertype), Legend originally had another problem. After its introduction, any person could play a Legend provided that that Legend wasn't already on the battlefield. If it was, that card was stuck in its owner's hand. They could cast it if they wanted to, but the newest one would immediately be put into the graveyard, so there was no incentive to do so.
This issue came to great prominence during the Masques block because the dominant deck was a deck known as the Rebel. The key card of the Rebel deck was a Lin Sivvi, Defiant Hero. The card was key to the deck so when two rebel decks played one another, the first person to get Lin Sivvi out had an unfair advantage.  However, developer Tom LaPille still preferred this version of the rule. 
From Champions of Kamigawa to Magic 2014
The second version of the rule used to affect the entire battlefield (regardless of the legendary permanents' controllers) and would send all of those legendary permanents to their owners' graveyards.  
- If an effect asks a player to choose a creature type, Legend is not a valid option.
- Changing a legendary creature's type or subtypes doesn't stop it from being legendary.
In other languages
- Matt Tabak. (May 23, 2013.) "Magic 2014 Core Set Rules Preview", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Sam Stoddard. (May 23, 2013.) "Legendary Rule Change", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Mark Rosewater. (May 09, 2011.) "The Issue Is Legen—Wait for It—Dary", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Tom LaPille. (May 13, 2011.) "A Legendary Disagreement", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Aaron Forsythe. (September 10, 2004.) "Legendary Rules Changes", Daily MTG, magicthegathering.com, Wizards of the Coast.
- Mark Rosewater. (October 04, 2004.) "Change For the Better", Daily MTG, magicthegathering.com, Wizards of the Coast.