Setting up Apple Mail to receive and send on a Google App email account is fairly straight forward. The Google Help Center has a page on this, and this is just a local mirror, perhaps with some additional points.
A quick description is - Open Apple Mail, create a new account (+), fill in the information, enable SSL on Incoming (POP) on the port 995, add an SSL SMTP server, on port 587, and on the 'Advanced' page slect password authentication, and it should be working. Now in more detail ...
1. Ensure POP email is ENABLED on your Google App account. This local page, or the Google Help Center page, have some help on doing that. POP is disabled by default.
2. Open Apple Mail, and click Mail, and select Preferences ...
3. Open the 'Accounts' tab, and click the plus sign (+) along the bottom to add a new account.
(a) 'pop.gmail.com' in the 'Incoming Mail Server:' field,
(b) your full email address (ex.: 'email@example.com') in the 'User Name:' field, and
(c) your password in the 'Password:' field.
5. Select 'Add Server...' from the pop-up labeled 'Outgoing Mail Server,' and enter 'smtp.gmail.com' in the 'Outgoing Mail Server:' field, and enter '587' in the 'Server port:' field. Check the box next to 'Use Secure Sockets Layer (SSL).' Select 'Password' next to 'Authentication:' and enter your full email address (ex.: 'firstname.lastname@example.org') and password. Click 'OK.'.
6. Click 'Advanced' and check the box next to 'Use SSL.' (The port changes to '995.')
Did you ensure POP email is ENABLED on your Google App account. This local page, or the Google Help Center page, have some help on doing that. POP is disabled by default. Now you should be ready to test the account. Ensure that it RECEIVES emails, and you are able to SEND emails.