This isn't an android thing at all... This is straight up a gmail thing.
This has happened to me though, and likely you've been hacked. You're also likely sending emails to all of your contact lists, and then they are immediately deleted and removed from your trash so you didn't even know they were there. So, there's a few things you need to do. FIRST AND FASTEST, you need to go in and change your password. Then, you ALSO need to knock off line who ever is online. They will remain online for a LONG LONG time if you don't remove them and end their session. Let me find how to do that....
Okay, go to
www.gmail.com then log in. with your email up, you need to go to the top right corner and find the Sprocket, click that, it opens a drop down box. Then click settings.
In settings, you need to scroll all the way down... ALL the way down. At the bottom of the settings page is "Save Changes" and "Cancel" buttons. BELOW THAT there is a Copyright for google. To the right of that copyright thing is a little thing that says "Last Account Activity: XXXXX" and has a number of hours. BELOW THAT there is a link that says DETAILS... Click details and it will bring up a new window. In that window, you will see "Access Type", "Location(IP Address)" and "Date/Time". There you can look through for any type of suspicious activity.
ALSO!!!!!!!! And most importantly, at this screen there is button that says "Sign Out All Other Sessions" HIT THAT BUTTON!!!! You'll have to log back in every time you go anywhere, but that's okay because you just reset your password anyway, right? RIGHT?! lol - just kidding... reset your password!
Get it done fast like and you'll be just fine.