The OS X 10.10.4 update includes a new version of the Mail app. One commonly reported problem after updating to OS X 10.10.4 is accounts stop working in the Mail app, especially Google email accounts.
The workaround that is helping a lot of people experiencing this issue is to turn on the option for “Automatically detect and maintain account settings” for the account(s) that aren’t working. This option is found within the Mail app by navigating to Preferences->Accounts->[account name]->Advanced. Quit and relaunch Mail for this change to take effect.
If this option is already turned on, try turning it off for the troublesome account, quit and relaunch Mail, turn it on, then quit and relaunch Mail again.