First of all, it is important to say that gaming industry is very perspective and has been developing rapidly throughout last 20 years. And , speaking frankly, most of the projects in this kind of business were failures. However there were also some very successful ones. So in this post I am going to focus on the successful ones by bringing them as examples.
The first model is a B2P model( buy to play). It is the simplest form of gaining profit just by getting customers to purchase a game. At the beginning of the gaming era the B2P model was obviously the most popular(not to say the only one possible) , however in the recent years in has fallen fallen behind a bit, while new models came into the market. The most successful games of this type were Cossacs, GTA series, WC3, Starcraft, Half-Life, Quake, CS, Civilization series, FIFA, Assasin's Creed and many others.
The second model is P2P model(pay to play). Honestly speaking, this model is rather unpopular, as people are rather greedy by their nature, so regular payments are not stimulating for gamers to keep on playing such a game. However, there were some games that succeeded with such model. Obviously, the first one that comes in my mind is Lineage 2 and its official servers. But it should be considered that the game was insanely popular at that time (and still remains popular enough) , so this model tend to bring success only if the game is really decent and unique by fitting the expectations of the players.
The third model is F2P model(free to play). This model is designed mainly for online multiplayers and is the most successful during the last couple of years. While reading this you might wonder what is the point of making a game for which nobody needs to pay to play - where is the money? The is not a single answer for this question .
Firstly, there are different F2P games. There are TF2P (totally free to play), PF2P(partially free to play) and P2W( pay to win) games. If we speak about the first category we mean games like DOTA 2 and Counter Strike. In such games the owner is making money either on so-called ''hats'' (ingame cosmetics and extra-features that do not directly affect the gameplay) or on additional activities, connected to the game( such as tournaments, advertising and etc). The idea is simple: people spend money to make the gaming process more comfortable and pleasant if they want to and can afford to, while those who does not want to do it are not forced to.
PF2P model can be compared to a trial periods of anti virus programmes or to free food lots in the supermarket: they give the player an opportunity to try a game while not giving him a full number of possibilities that the game provides. This model is easy to understand , but not as easy to bring into life, cause such limitations ought not to impact the gaming process too much. DC Universe online is the game that suits this model the most.
P2W model is also very popular nowadays and is used mostly in online multiplayers too. This model's idea appeals to economic game theory. The whole picture is like this: there are some features or extra values that could be bought for real money and are either inaccessible for those who don't want to pay or very hard to earn/reach. So, bringing up the game theory we assume that the most logic and right behaviour for the whole group is not to get them , so that everybody could not pay and stay in comfortable circumstances. But as we know from the game theory (or our life experience) individuals will choose the strategy which is individually better , so they will pay the money. For example, games that run this model are LOL(league of legends) and WOT(world of tanks).
I would like to stop a bit more on this last model as it is not as obvious as it seems. First of all, sometimes the games declare being F2P while they actually P2W( WOT is an excellent example), as it is easier to get a new player into the game and later force him to pay the money instead of trying to force him already knowing that he would be forced to do it . Secondly, sometimes P2W does not affect the game balance seriously( for example - WOT until the ''economic policy changes'' ) , while in other cases it just makes the game really hard to play without donating into it(WOT after Silver Gold patches, when the balance has just crushed and did not return even in years), so the game-developers should pay a lot of attention towards this question.
Finally, it will be fair to say that there are a lot of games that mix different types of the models or bounce between some of them. For example, Hearthstone which came into market recently is a mix of TF2P and P2W models, as it is much easier to play donating, but also absolutely possible without.
All in all, the game industry is developing very fast and new models are probably going to come up soon, but these ones are the most general for today.
I hope i covered this interesting and actual issue as much as i could. Feel free to disagree or criticise in commentaries.