All of your information is stored in databases.  Once the integrity of that database is compromised, either by malicious means or by accident, this it is considered a breach.  Even if you do not have an account on line, that database still exists.  An on-line account only means you now can access your data over the internet.  

So how does a breach affect you?  If you never had a account with that vendor, then it probably doesn't affect you.  However, a data breach at a company that holds records of people whether they have business with them or not, such as one of the credit providers, can have a significant impact.  Depending upon what data was stolen or released, it could be the key to stealing your identity, opening accounts in your name, and ruining your credit.  

If you receive a letter from a vendor advising you to change your password, do it.  However, one of those pieces of stolen information may be your email address.  BEWARE the scammer who send you a fake email with a link to change your password or update your account!  If you hear about a breach and think you may have data there, log into your account (NOT from a link), and change your password.  

FREEZE your credit files.  This will prevent a criminal who does receive this data from opening accounts in your name.  And it is now free to do.  Also, CHECK your credit report annually.  If you are in Georgia, you can check each of the three credit reports annually for free.  You can stagger them - check one, then four months later, then another.  Four months later check the third.  The three are Experian, Transunion and Equifax.  Clark Howard has more information on his web site - Credit Freeze Guide.