What has been posted is very helpful. But some minor points....
Im not sure that pbem is "most games" altho it might work if it was said as "most server hosted games in this forum". Even then Im not sure. Im fairly sure that most of the games done by small groups or from the various irc channels are direct connect. And my server, Dom3Minions, has never been busier (its almost entirely direct connect games). Both for the continually open games I host, and for special setup games such as you see in the forum here. Altho most of the special setup games on dom3minions arent from this forum since apparently there is no need for them here.
Personally I consider pbem to be a headache both to do and to run. I had given up on running them long ago setting up Dom3minions for direct connect, and was thrilled later when LLamabeast started his server to support pbem. People also like it because it has a menu of options for starting a game. I did a menu like that long long ago also but it doesnt work as well for direct connect games so now I just leave the options completely open to the person creating the game.
Playing direct connect is pretty simple. You use the Network option on the menu to connect, and then it plays like a solo game. You do need to know how to turn on mods (if that game wants to use some), create your god ahead of time with a password and save it. But you need to do that whichever method you choose to play multiplayer games.
Some like pbem for the "automatic backup" feature of having their turns in their email. But actually many direct connect games offer that also. Its very simple to have turns emailed out. Its just receiving player turns at the server that is a headache. Or they like it for the record keeping and admin functions they see implemented on llamaserver (altho those also are available or unnecessary for direct connect games)
If you want to try out both types for yourself before diving into multiplayer games then feel free. Here is a list of games that are always available on my system (a new one created when one of those starts). Pick one then read the How-To link. Id recommend playing a vsAI for this purpose since you can start it with just one player. If you get tired of it then just set yourself to computer player which will tell the game you are done and save me some cleanup.
http://www.dom3minions.com/HostedGames.htm
I suppose you can do the same on LLamaserver for pbem by creating a game and joining it twice but I cant speak for whether or not LLamabeast would like people doing that.