I have created a new user with "useradd" than changed the password with "passwd" (logged in as an administrator). After that I tried to login as the new user but couldn't. Error message was: "call to lnusertemp failed (temporary directories full?). Check your installation".

I currently use Debian 8.

I checked the home directory and couldn't find a new created folder for the new user (wasn't supposed to be done automatically by useradd?) and I checked the etc/password and there was the new user name inside it though.

What should I do? I really need to have more than one user on this computer...

There's more to creating a user than just creating a login name and password.

My suggestion would be to delete that user, than recreate it with adduser, which is a much more robust command. This article gives a good comparison of the two.