You don't need any tools at all to change strings, but there are a few tools which will make it a lot easier to change your strings.

String Winder

A string winder is a real handy tool to have around when you're changing your strings, especially if you're changing all of them at the same time. They really save a lot of time and energy and I don't think you should be without one! There are electric winders available but they're a waste of money and more likely to be dust gathering clutter than a useful tool.

Justin Recommends: D'Addario Pro Winder (it also has string cutters built into the handle!)

String Cutters (or Pliers)

For many years I used a regular pair of pliers to cut my strings, until I got a winder with the cutters built in! But any kind of wire cutter will do the trick for sure. One advantage of pliers is that you can bend the tip of the string end over so you can't stab yourself with it!

Justin Recommends: D'Addario Pro Winder (string cutters in the handle!)

Headstand

I used to change strings sitting in a chair but recently got into doing it on a table or the floor and if you're going to change the strings with the guitar lying flat you'll find it a load easier with one of these headstands. Pro guitar techs usually have a kind of crade for the neck made of wood with foam on top but these little folding ones are fine for us regular folk not doing it every day!

Justin Recommends: D'Addario Headstand

Polish Cloth

The best time to give your guitar a clean is when you've got all the strings off, so grab yourself a polish cloth and have it handy when you do your string change! It's also handy to rest the body of your guitar if you're doing it on anything but carpet!

Justin Recommends: Any old t-shirt!