I saw some YouTube videos with people playing champions before they were released. How can I do that too?
Before September 2012, PBE signups were open to all summoners.
From September 2012 to December 2012, PBE signups were open on the 1st, 2nd, and 3rd of each month.
From January 2013 onward, Riot decided not to hold scheduled PBE signups, due to the large amount of players.
As of this writing (October 2013), PBE signups are open, ostensibly to prepare to test Season 4 features.
Additionally, the PBE structure has changed. PBE will now have registrations in rounds, and the majority of accounts will be disabled at the end of each round. (There is no longer permanent access to PBE.) (source)
There are also new restrictions on who is able to create a PBE account:
It's also worthwhile to note that PBE can no longer be signed up for per se; rather, Riot will select from a pool of applicants. For instance, players who tend to play Mages are more likely to be selected when testing a new Mage.
This appears to be the url to sign up for access to the public beta server:
The beta server is only NA, so there is no EUW etc. equivalent. As mentioned by Schism, it is no longer permanent access and you need to apply each round to get in. Riot has stated that you may be more likely to get access based on contributions (e.g. bug reports) during past rounds in PBE.
You can only sign up for the Public Beta Environment on the 1st, 2nd, or 3rd of each month.
See this thread for more information: http://na.leagueoflegends.com/board/showthread.php?p=28042411