What browser are you using? As already mentioned by cj, I do believe you must have it set up to save cookies.
I use firefox, and if you go to the top menu bar and ckick Tools, then click "options/security", you will see in the middle of the page where you can choose to save passwords.
Any changes you make to your browser configuration often do not take effect until you exit your browser and restart it. It's a good idea to do that right away when you have everything set the way you want it.
When you first go to a site that requires a username and password you will have to enter that information. Then the browser will open a window and ask if you want to save the password. This will happen as soon as you enter the password. Watch for it, because it doesn't always stand out when it comes up.
I'm sure that other browsers have the same features but you will have to explore a little to find out how to configure your options.