Generally, you can use any kind of paper to print printables, but it does depend on the type of card and how 'fancy' you want your printable card, invitation or paper good to turn out. Below are some guidelines. I use 5 types of paper for printing printables:
Regular ol' printer paper - All product plugging aside, the 'fold only' line of e.m.papers printables were pretty much designed to use with regular weight printer paper.
Plain stationery Paper - If you want your 'fold only' printable to look a bit more spiffy you can also use plain stationery paper, the kind you can buy by the (small) box and would print out a resume on. You can get usually get this type of paper at a stationery or office supply store. I'm a 'quick and dirty' kind of gal, so I always keep a box of this type of paper around around. The ability to print out a nice card, fold it, sign it and stick it in an envelope 4 minutes before we leave for a birthday party is a convenience I'm starting to become heavily reliant on.
Custom or specialty stationery paper - Fancy 'pay by the sheet paper'. The kind of paper, is similar in weight to regular printer paper but found in stationery, craft and art supply stores. Usually in unique colors or treatments such as slight transparency (e.g.vellum). You can do a lot of cool stuff with this paper, however it's not cheap. My rule of thumb is to purchase a sheet or two, experiment to see how it works and then purchase more if I like the results. Below is an example of a really nice ivory colored, richly textured custom paper I purchased to make an wedding anniversary card for my (soon to be) in-laws. I was really pleased with the results (this is now the basis design for one of our 'Monogram' printable wedding invitation set):
Card stock - This works best for paper good that you want to have a little stiffness and weight (cut and fold greeting cards, birthday cards, postcards bookmarks, recipe cards, etc.) There are a multitude of card stocks available anywhere between 60-127 lb. weights. The thickness increases with the pound size. How well each one works depends on your printer. In spite of the fact that I'm running a printable business, I actually use a really old HP Deskjet 960c that I work to death. It handles 110lb card stock without a problem. This weight works really well for greeting cards and is about the thickness of a report cover or business card. Depending on the brand/type, card stock does seem to suck up a lot of ink, I'm still experimenting with finding the best brand.
Photo Paper - This can be a great choice especially when printing cards with large fields of color. Colors appear more vibrant and smooth. The same rule regarding paper weight (higher the pound number, the thicker the paper) applies. Below are two examples of our 'SF Valentine' card, one on photo paper, the second on regular card stock.