I keep getting messages that my sign in to my gmail account has failed. I am not trying to get into my account. I did go in and change my password this morning because of that. What do I do now? I would guess it has been at lest 10 times.
Gmail has a helpful feature called "Last Account Activity" that shows you what has been happening with your account. It's found on the bottom right of the gmail window.
Full details are at https://support.google.com/mail/answer/45938?hl=en.
With this, you can get a rough idea of where the activity is coming from. This will help rule out two possible issues: your little brother (or wife, or room mate, or whoever) trying to log on to your computer when you're not there, or a old machine of yours somewhere that is trying an old password.
Beyond that, if some anonymous person on the Internet is trying to log onto your mail, there isn't much you can do to stop them trying.
But there are things you can do to make sure your gmail account is even more strongly protected than it is now. Even though they are failing now, it's always wrth looking to see if there is more you can do.
Google have a checklist of things to do to "harden" GMail at https://support.google.com/mail/checklist/2986618?hl=en&ref_topic=1669043.
Four of their suggestions are particularly important:
This last option is very applicable to your problem. With 2-step enabled, an attacker will need both your password and access to your mobile phone in order to log on as you.
Somebody can try to login to your account as many times as they like. Ensure you have a strong password and that's about all you can do to protect yourself. The fact they know your email address doesn't really matter and if your password is strong, there is no problem.
You could also setup 2 factor authentication to further protect yourself: http://scotthel.me/PasswordSecurity