Windows and Linux for sure, and I'm pretty sure they have an OS X version too.
The "Core" basically runs on a server and connects to your IRC networks and channels, and then your client connects to your Core server and syncs chat. While your client is closed or your computer is off/sleeping, you usually appear away, but the core stores all chat in your joined channels. When you open up your client again, your user experience is the same as any IRC client, but you get scrollback for however many lines you set (I think I have around 10,000).
The only issue with the core is that IRC has such heavy disk I/O that it can slow down your server if you're not careful (I'm joined to about 30 very active channels though). Also many server hosts specifically forbid IRC-related servers for that reason.
Edit: there's also an Android client as well, which is pretty nice.