any type of paper can be folded successfully. The
important thing is to choose paper of the correct size,
quality and decorative appearance for the design you
intend to fold. This is a matter of experience, so,
before you invest in expensive papers you may never use,
try folding a few simple designs from ordinary paper
You will however need to know whether the design you intend to fold requires paper that is the same colour on both sides (homogeneous paper) or whether it should be white one side and coloured/decorated the other (differentiated paper). The diagrams you are folding from should tell you which of these two kinds of paper you require.
Ordinary photocopy paper is a good source of homogeneous paper.
You can buy differentiated paper in packs of brightly coloured squares from many craft retailers or from the Muji shops run by Liberty's. Differentiated paper is sometimes known as kami (which just means paper in Japanese) or irogami (my own preference) or sometimes just 'origami paper.'
You can also do origami with 'found' papers such as brochures, flyers, unused wrapping paper etc.
Occasionally you may also come across designs that need to be made from special types of paper or even card (card is just thick paper). Dollar Bill folding and Business Card folding are both popular in the USA. Metro-ticket folding is popular in France, and Teabag folding in Holland. (Actually it's the teabag labels that you fold rather than the teabags themselves. Even in Holland, they still make tea with the bags.)
The important thing about any paper suitable for folding dry (the normal way of doing things) is that it takes a sharp crease well. If the paper won't hold a sharp crease or splits along the folded edge so that the colour cracks then don't use it. Look for something better quality instead.
If on the other hand you are going to fold your paper damp you will need to look for another type of paper entirely. There is plenty of advice on the web about this speciality.