1. Buy to Play: Think 'regular video game'. The developers can later make "expansion" packs for the game and sell that to get money too. (sadly, the introduction of internet shopping has made too easy for the developers to rip people off by supposedly selling them a whole product for one purchase while withholding already developed content for the game and selling them off as "DLC". This not only reduces the value of the game but also inflates it's price buy raising the cost of obtaining all the game's content (ie, Skylanders, Mass Effect).)
2: Pay to Play: Pretty easy to figure this one out. Basically, in return for high-quality game content and patches being developed so that your game can be funner, the devs ask their customers to pay a monthly fee (usually between $5 and $15). Sadly, very very few MMOs nowadays offer high enough quality content to justify the subscription cost, and so therefore they switch to the later mentioned Free-to-Play format to recoup their investment.
3. Play to Pay: This is Pay-to-Play with a twist: players can use money they acquire ingame to pay for their subscription fees (EVE Online, Wildstar). Unfortunately, this means that a player would have to play the game very regularly (ie, daily basis) to obtain the money required to bypass having to buy for subs with real money.
4. Free-to-Play: Alright, the games using this format aren't really free, but it's close enough. This payment method is where players can simply go to their website, make an account, download the game, and start playing. The catch? Devs try to squeeze money out of the players by producing special, tempting content that can only be acquired by paying real money (ie, nice looking spaceships, mech paint, etc.). The other downside is that it makes it too easy for the developers to focus all their time and effort on paid content rather than work on the "free" content (Rift, SWTOR).