Your account number identifies your specific checking account at your bank. Unlike the routing number, which is shared by every account at that institution, your account number is unique to you.
It's printed in the MICR line at the bottom of a check, usually right after the routing number and before the check number. Account numbers can range from about 8 to 17 digits depending on the bank.
Treat your account number the way you'd treat a credit card number — it's printed on every check you write, so avoid sharing images of checks publicly, and consider printing your own checks on demand rather than keeping a large stack of pre-printed ones with your account number sitting in a drawer.