Keychain passwords suggestions that I can't remember, how do I log in from a different pc I hope that I put it right, that you guys understand. 

Everywhere I read that Keychain can suggest strong passwords and remember them so I don't have to remember them. Works wonderful.

The thing is, I am scared to do so, because; what will happen if I have to enter one of these sites from a different computer. 

Say that I let Keychain come up with a great, strong password for twitter. 
Next time I log on to twitter from an unknown computer, can I still log in with my old password? Or do I actually have to remember the password that Keychain came up with?

The logic thing would be that it actually changes that password, so when I need to enter that website from a different computer, I will still have to use that hard to remember password. Which means I need to keep an extra passwords list somewhere accessible. 

I've been looking all over, but I can't seem to find it, so I figured, I'd ask you people.

Can you help me out?
While Keychain Access and the iCloud Keychain is specific to the Mac and iOS, there are other applications such as 1Password and LastPass which have various syncing options (Windows, iOS, Android, Linux, and even web-based) so you would never be without access to your passwords.