This isn't true paper cutting, but something I indulge in ever year. I love making paper snowflakes, and in the UK they often serve as a very nice White Christmas substitute. This site has some great templates, you just print them out, fold as instructed, then cut out all the grey areas. If you want a slightly more 3D version then these are pretty.
If you are a bit geeky then I think these might just be the best thing since sliced bread... I may set the entire family to the task of making one of every kind on Christmas Day, whilst watching Star Wars.
On a slightly more technical level, there's also proper paper cutting. Unlike the snowflakes you do need more than a sharp pair of scissors...
I love the graphic simplisity of this craft.
A kit is a nice way to get started, as that way you get a proper cutting mat, a scalpel that is nice and sharp, and some beginner templates.
If you want to go freestyle then this site has some nice beginner templates, and hints about how to make cuts. This site has information about how to create your own papercut from an image you've picked yourself.
This post is part of my advent calendar 2016. Every day in December we're saying so long to 2016 by embracing learning, knowledge and creativity. Each day there will be a new post with a new skill or craft. You can read all the past posts here.