Apple Mail for OSX needs to be configured to work properly with email (and Mailbox). Here's a quick how-to:
You've likely already done this, but if not, start by adding your Gmail, Google Apps, Yahoo, or iCloud account to Apple Mail:
As of version 10.8 (Mountain Lion), Apple Mail gives you the option of also setting up Google Calendar, Google Talk (messages), and Notes.
If you’re on Mavericks, make sure Mail.app is updated to the most recent version.
Once the account is created, you'll likely want to get junk mail out of your inbox. Do this by going to Mail > Preferences… > Junk Mail and selecting "Move it to the Junk mailbox"
OK. Your basic account is set up! But if you stop here you'll notice all sorts of weird issues: search results will include oodles of draft messages, messages you send from the web or your phone won't appear in your "sent" folder, and other similar issues. This is because the folders Gmail, Yahoo, and iCloud use for Sent, Drafts, Trash, and Junk by default are different than the ones used by Apple Mail by default. Here's how you fix that:
If you look inside the [Gmail] folder on your left navigation panel you should see a number of folders including "Drafts", "Sent Mail", "Spam", and "Trash". If you can't see this panel, hit the the"show" button in top left of your app:
We need to map each of these folders to the ones Apple Mail uses for Drafts, Sent, etc. Do this by selecting one folder (let's start with Drafts), and then in the menu selecting Mailbox > Use this Mailbox for > Drafts
Great. Now, repeat this step for Sent, Trash, and Junk (use Spam in [Gmail] for Junk).
If you've been using Gmail through Mail.app for a while you'll probably find new folders in your sidebar but outside of the [gmail] parent folder. They'll have names like "Deleted Messages", "Sent Messages", "Drafts", and "Junk". These are the old folders that Apple Mail was using to store drafts, sent messages, etc. Most or all of the items in these folders will be unnecessary. Your new (real) "sent" folder already has a copy of every sent message, for instance. Junk is probably full of junk and drafts probably full of old drafts. Trash is trash. You can leave these folders alone if you want, but you can also select all items in them and delete them, as well as deleting the folders themselves. You won't be using them anymore.