A database is a collection of data stored usually in 2D arrays aka tables. On the web specifically the login and register forms we see most commonly 2 input fields. First is the email and second a password. Obviously they are both very secured to prevent any SQL injection attacks and are also checked if the data entered perfects the criteria. Usually your email has to exists to be considered valid otherwise you end up with a certain error telling you to enter a valid email. After you enter your email you go to password field... here the text is usually hidden with black dots while you type it for security reason a password must also contain lowercase , uppercase, numbers and symbols so the program checks them all asynchronous just like with email.
The final step check if email and password are valid and if so you may proceed to click a button register. When this button is clicked there are couple of things you may assume happens. First email and password are written into a DB table and assigned a unique identity. Then your password gets encrypted with SHA256 or MD5 and others. The password stored in DB is always of encrypted form so database admins don't know your password they just know your hash.
Your hash is an encrypted password and when you go to login page and login with information you registered the password you enter is converted to hash and then compared with the hash of the password their DB. If it's a match you are able to proceed.
Hope this helps.
r3t40