First things first. Do you know what is broken? Is it the cigarette lighter (bit you push in and then pops out to light 'yer fags') or the socket (hole in the dash).
Test the socket by using a 12v appliance such as a phone charger, cool box etc. Remember to have the ignition on just in case it is only live when the ignition is on.
Assuming that your phone charger etc works, then the new cigarette lighter should work in the socket.
If the socket is not working, I would be surprised if it was actually faulty as it is only a couple of contacts in a housing. First thing is to check the fuse (look in the owners manual for fuse location). If it is blown, replace it and see if it blows again, chances are it will be OK with a new fuse.
If the fuse is OK, then it may well be the socket itself. Sorry but I have no idea how to remove it on a '03 Clio, but it shouldn't take much to figure out.
You will need to somehow remove whatever part of the dash the socket is in. It may well just unclip, or you may have to remove a screw or two. Once you can get to the back of the socket you will be able to see how to remove the socket. Most likely it will be a couple of clips on the side of the socket that you will be able to release.
I'm sure the termainals on the back of the socket will have push fit connectors, just pull them off of the old socket and push them on to the new one.
Very important: Make sure you note which wire goes on which terminal and put them on the new socket in the correct place!!
Hope this helps and good luck!