There are so many different types of paper, it can be a confusing choice, but here are some places to start.
Origami paper (Kami)
Perfect for simple models. It comes in a huge variety of colour and pattern combinations, is usually very square and often very cheap. Each pack will contain a selection of colours, although you can buy packs of a single colour (such as red, when Christmas approaches). It has a colour/pattern on one side and plain white on the other.
Duo (double-sided) Paper
As above, but both sides of the paper have a different colour or pattern on each side.
Harmony Paper
Another type of origami paper but with patterns made by fading colours into each other. It's perfect for flowers and decorative models.
Kraft Paper
Alias wrapping paper, the sort that is generally brown and comes in huge rolls. It’s brilliant for folding and you can also find it with different colours, if you look around. The manufacturing process entails treatment of wood chips with a mixture of sodium hydroxide and sodium sulfide, so it’s not a good idea to make your own.
Elephant Hide
A type of paper with an elegant marbled effect, that is also good for wet-folding. Not literally made from elephants, you’ll be glad to hear. Also known as Wyndstone paper.
Washi
This is a Japanese word meaning “hand made” paper. It is generally thicker than normal paper and requires some experience to fold with, but can produce outstanding results that last for many years.
Damul / Papier D'amour
This is thin paper (50gsm) and is aimed at more experienced folders (though this is not to say beginners can’t try it as well!) It is printed on French Kraft paper with a soft matt surface on one side and a subtle gloss on the other. It is quite suitable for more complex folds.