For many hundreds of years, perhaps since the early invention of paper, people have been folding it into simple shapes. Back in the eighteenth century it was a popular art form in Japan and more recently Yoshizawa created thousands of new origami designs.
More recently with the introduction of mathematical principals into origami, it has become possible to represent more complex shapes in finer detail.
Often a simple flat surface and your hands are the only tools you need. However, you could choose to purchase, bone folders to be able to create sharper creases, teasers to make small folds, paperclips to help hold folds, a ruler and embossed to mark out lines. A water spray is handy for wet folding.
Origami paper, which is perfectly square. The paper comes in many different sizes and colours, often white on one side.. Washi is one of the most popular origami paper but there are others to choose from, like foils that you would need to make by hand.
The rules that make a piece of work origami are that there are no cuts (only folds) and that it is from a single square piece of paper.
You can print out a crease pattern that will help you create shapes. These crease patterns follow four basic rules:
1. You can colour a crease pattern with just two colours and the two colour will not meet
2. Mountain and valley folds always differ by two
3. Odd and even angles always add up to a straight line.
4. A sheep can never penetrate a fold
Once you know the rules you can apply them to any piece of paper and start to create your own designs.
In the 1990s, people starting looking a the way you pack circles into a square, as each flap created needs a half circle of paper to create it.