You are getting there, lesleyhannah.
On my router the yellow cable does the same as on yours and connects the computer to the router.
The thin black wire goes through a transformer into the wall plug for the electricity to power it
I don't have a blue cable because my modem and router are in the same box.
While waiting for detailed instructions on what to do you could try and have a little look yourself at the router settings.
On my computer I type
http://192.168.2.1/ into the address box at the top of internet explorer and hitting enter takes me to the router setup utility. I have heard that other routers use the same address code. It would be your modem you are looking at however, not someone elses.
At this stage I login without writing in a password, not needed to look at your settings.
Hopefully you will see amongst a lot of other information a line headed SSID. If there is a name or sequence of letters/numbers written beside that then you are looking at the name of your router. You can change this name of course.
If you do find your SSID, turn on your laptop and ask it to find a wireless connection. It should find one with that name. If it does then accept it and try to connect to the internet. If you have no password protection on the SSID then you should connect without any problem.
Back to the router, somewhere there should be a menu item with the word security. Click on that and you will be taken to a page where you can enter different key numbers. I have key1 set to my password.
If you see these different things then come out of the router web page (by clicking on the back arrow or closing the tab/page) without changing anything and report back here, saying what you have found.
see you later