I have one domain for which I use Google Apps (Standard Edition) to provide email. I've recently started another website at another domain, and I want to also use Google Apps (again, Standard Edition) to provide the email for this new domain. Do I need to go through the entire registration process again and create a whole new Google Apps account for the new domain, or can/should I add it on to my existing services?
Short answer: Yes.
However, there are some things I think I should say about that approach.
For completely unrelated sites, it's probably best to sign up for completely unrelated Google Apps accounts. Domain aliases will work, but all sites will share the same limitations on how many users can be created (quotas). Separate accounts will each have their own independent quotas.
If you alias, you'll have 50 free users to divide among all the domains. If you create separate accounts, you'll have 50 free users per domain.
Hope that helps you. Personally, I'd create a new account.
I have Google Apps Standard Edition, and the option for adding a domain alias is enabled for me. I think it's exactly what you want to do: