Frames. Iframes is another one, but there's still a good number of browsers that won't support it.
Or you work out with the other website about hosting issues. Does the other web site host your entire application to be run on their server and site? Or do they give up and redirect their users to your site? Having them come to your web site is what's called ASP or Application Service Provider. The first method is more like selling your web app and the rights to use it to them.
If you don't want to go that route, then you will probably want to get into XML and RPC (remote procedure calls). You'd have to create documention on how to interact with your service and have the other web site create web pages (and code) that interact with your web pages and code. This is probably the ultimate in control, but requires a good bit of work on both parties.
