A "compromise" means that your account name and password have been discovered by someone other than you. Often an attacker will change the password of an account they have compromised in order to prevent the owner from reclaiming it. From there, the account can then be used by the attacker for various purposes depending on the type of account that was compromised. For email accounts, this often includes sending phishing messages to other users, stealing information found within email, or filing password reset requests with other accounts that list your email in order to steal additional information, such as banks, credit cards or social media.
There are many ways you can improve your security and stay safe online.