Basically what QM said holds true. Online Gaming and Internet Gaming is one and the same thing.
MMORPGs or Maasively Multiplayer Online Role Playing Games are games that are based in a virtual online world. At any given point of time, for a sufficently successful MMORPG, there will be a few hundred thousand players playing the game and all of them inhabit the same online virtual world. These games never end. You as a player, inhabit the virtual world, carry out the tasks that are required in the game and proceed further. If you are to logout at any point of time, the game wont stop...it would continue like it should just like the world around you doesnt stop when you go to sleep. Such games typically include levelling up, virtual money, quests etc. You are mostly required to pay a monthly fees to be able to play the game besides the cost of owning the game. There are some free MMORPGs available as well.
Then there are the games that have the multiplayer feature. For this, you need to login in to the game servers, find out other people who have been playing the game online, and then join the game. The number of people playing the game online in one sessions depends upon the number of gamers that the game can support i.e. it can range from 8 players to 64 or maybe even more. However, your virtual arena is limited to the server thats hosting it. There will be tons of similar arenas with similar or completely different rules.
Another aspect of online gaming can be found in Co-Op gaming where you teem up with your friends online to play through single player levels as a team. For example, you can play through the S.W.A.T. 4 levels as an actual SWAT team, with your friends controlling the other SWAT members that would have been controlled by the A.I. had you been playing alone.