When I open up a terminal and write
# mozilla
for example, then mozilla starts but I can not continue using the terminal because it's making some outputs on what mozilla is doing, how can I open a program using the terminal without having it 'stuck' to the terminal?

type for instance "mozilla &"

try

[user@domain]$ mozilla > mozilla.out &

Okay cool, works.

But is it not possible to remove complete contact with the terminal, like if I use this when I close the terminal mozilla will also close, it would be nice if it where possible to open a program using the terminal but once I close the terminal the program I opend will not.

Well I can just use the 'Run' instead I guess.

as far as i've seen backgrounding the process (via the &) severs all ties to the console. Try it out and see what happens

try killing the programs via the terminal
ie:

user@domain]$ ps
5102 mozilla
user@domain]$ kill -9 5102

Actually I just learned the solution to this myself. If you start mozilla like this:
# mozilla &
the program will start up correctly and you will still have control over the command prompt.
But, if you wish to close the command window while still using the program, you have to type:
# exit
at the command line and the command window will disappear while leaving the program running.
Simply clicking the "x" on the console won't do the trick because it's like prematurely ending the process. Typing exit causes the command window to close properly which leaves background processes running.