Web browsers like Chrome and Safari have built-in features for managing passwords. These features can be set to remember and auto-fill passwords. Instructions for managing your password settings in Chrome can be found here, and for Safari here. If you’re looking for a more secure way to manage your passwords, I would recommend installing a password manager.
A password manager uses one master password to manage all of your passwords and personal information (like website logins, credit cards, banks accounts, passports, drivers license, etc.). The built-in security features for a password manager can detect when someone is logging in from an unauthorized device and will send a verification email to the address on your account before it grants access. The other major bonus of using a password manager is that it can randomly generate passwords for you.
If you decide that you don’t need this much security you can keep a little black book of passwords hidden somewhere safe, away from your computer.