My computer system was set up by a knowledgable friend.I know very little about the technical side of its working.
I have broadband and am connected through a router.
My wife would like to buy an Apple IPad but have been told in the shop that she will need a Wi-Fi internet connection to send e-mails etc.
I do not know if my system uses Dial Up or Wi-Fi - how can I tell?
Does Wi-Fi need extra equipment and if so what?
Dial-up and WI-Fi are completely different things. You connect to the internet with broadband therefore you do not have dial-up.
If your computer IS NOT connected to your router with a cable then you have Wi-Fi.
If your computer IS connected to your router with a cable then you may or may not have Wi-Fi. In this case does your router have little aerial(s) sticking out of the back? If it does it's Wi-Fi. Either Google your router to check if it is Wi-Fi or post the make and model here and somebody will have a look for you.
As Marvin42 says, you do have Wi-FI but it may not be turned on/set up?
You will have to follow the instructions in the manual you have? It will tell you how to access the router set up menu, which is typically done by typing 192.168.0.0 or 192.168.0.1 or something similar in to the address bar, you will then be asked for a password, again typically it could be "admin" "admin" or "password" "password" the manual will tell you what it is, this will get you in to the set up pages for the router then you can follow the instructions in the manual to set up the wi-fi.