As a primary strategy gamer, I can say that online play is an entirely different experience. You meet a few greifers, a few cheaters, but its not as bad as some people make it out to be in that respect.
When I played say, company of heroes, playing online isn't something I did "just for fun". I did it for fun, yes, but it wasn't a relaxing space between work. It was something I dedicated to for that time span, something I thought about and worked at. It was more like a hobby then entertainment. It was a challenge, but it was fun.
The appeal, in that type of game (strategy), should be a little more obvious then in some other genres of games- playing against a thinking human being that reacts and predicts your movements and plans is something else entirely from a computer player with limited strategic options but more resources to play with (For those not into RTS, that's generally the norm- a computer player gets more units or money or strength to compensate for the fact it can't actually think). So now I am predicting what my opponent will do, baiting him into making a mistake. Is he falling for it? Or is he leading me on? Company of Heroes in particular I felt epitomized this facet of the RTS genre, especially before the expansion pack came out. Those are questions I asked on a regular basis throughout a game, matches were close and fast-paced, and who was smarter and could better think things through seemed to have a much greater impact on the game then other RTSs (say, starcraft) where the speed at which you click and type is the determining factor.
As for the two major types of online games- MMORPGs and FPS- there are different reasons to play each. Though I think the primary reason is social.
Playing an FPS online- say, Teamfortress 2- is something a bit less strategic. My opponents and allies are just characters in the game, they may as well be computers. Granted there is something different that just can't really be explained about being killed by, or killing, a player I know is actually a
player, but honestly the gameplay itself isn't so different as playing against what could be really well-done computer players.
But what I play it for, and a lot of other people, is the social aspect. Its like meeting people at a party- you and a bunch of strangers come together to share a mutual interest (the game), to work together/compete with eachother. It isn't hard to find a group of people who are just trying to have fun because, well, its a game, people are trying to have fun. There are people who play competitively, as well, although I don't consider myself one in that respect (I am or was really good at a couple of the games I've played, and I play to win as hard as I can, but I do it for fun, not to beat the other team) understanding that motivation should be easy, just look at team sports.
We've had a lot of crazy and memorable conversations during a lot of the games I've played, and I've met people who I've played games with on a regular basis for a while (No-one I would hang out with in real life, but the idea of having a separate digital and "analog" life I think is becoming more and more true in this day and age).
On top of that a big one for me is the feeling of working togetether towards a common goal, even if I am working with strangers. I'm certainly a co-operative person, and it really comes out in gameplay. Capture the Flag Unreal Tournament, for example, is something I've had a lot of fun with, along with Battlefield 2142, in which I was a regular squad leader. As stated even if my opponents were just extremely difficult, well-designed AIs, working together with people, communicating and planning, its a unique experience. Even in a loss, the sense of accomplishment is great, and the fight itself is an entertaining challenge to work against.
As for MMORPGs: I prefer the single player variant. Its the story and characters that drive me on in those types of games... MMOs lack both. There can be cool
looking characters, but beyond that, its just your imagination.
The thing is finding something right for the person. Something the industry is right now, working very hard to accommodate. Some people are intimidated by the (assumedly) competitive nature of online gaming, others are simply awkward or don't want to be the "new guy", and other think of it as work, a hassle, a chore to get into with no real payoff. And the truth is for people who aren't into it,
getting into it can be a trying experience. The biggest challenge is trying to find which type of game you'll enjoy online, something most people don't consider properly. But I think That'll be for another post