Nowdays, use Discord.
While this question was already answered nearly ten years ago, I want to give a more up-to-date answer.
In 2021, Discord is a good choice for play-by-mail or non-realtime roleplaying games. Several previous contenders have been discontinued (Google Wave in 2012, Apache Wave in 2018, Yahoo Groups 2020), and Discord has a number of advantages over existing options.
While Discord resembles IRC in that it's a text-based chat system, a critical difference is that it retains chat server-side, allowing it to be used for non-realtime communications.
Discord is free, and allows you to set up a private invite-only server for your gaming group. It allows multiple chat channels and direct messaging. It can integrate dice bots and other roleplaying game support bots. It's not run by Google, who have a habit of starting new and innovative communications apps and shutting them down later, so there's a good chance Discord will still be around five years from now.
It is web-based, and also has well-established apps for iOS, Android, Windows, Mac and Linux, which nowadays allows people to receive notifications of updates and play on the go. You can't generally do this with a web forum. It's very useful for the DM to enable mobile notifications and respond quickly, as the DM can be the bottleneck in many systems. Servers are also run by Discord themselves, which makes it more reliable than many fan-operated internet forums, which are at greater risk of shutting down due to technical malfunction or other problems.
In my experience, play-by-post is much slower than in-person play, and you have to make adjustments for this. The main things you can do to mitigate this:
- Pay attention to bottlenecks which block play. For example, the standard initiative rules in D&D would mean that all players wait on one player. The DM is often also the bottleneck, so I have used a "players make all the rolls" variant so that combat can continue without me.
- The strength of non-realtime is that you can type lengthier paragraphs, take more time considering strategic decisions, and divide the party more easily. Consider using systems, house rules, or modes of play which play to these strengths.
- In a game like D&D being run entirely on in non-realtime mode, use milestone XP. You won't get through as many combats in the same space of time.
Read the play-by-post and online-roleplaying tags for further questions discussing this topic.