Finish: Your paper finish will affect how you're able to print and how much ink your invitation will take. Glossy papers take a lot more ink and longer to dry. Linen papers and papers with more texture have a much higher tendency to bleed, and so they won't work with some at_home printers where you can't change your ink settings. If you're not interested in researching finishes because you're not a weird nerd (like myself) who likes to feel on paper, a matte card stock is an easy, relatively foolproof option that will more than likely work for the printer you're using.
If you're buying your own paper, you can find a good selection at local office shops when it comes to card stocks and glossy photo paper. However, if you're looking for linens or specialty papers, you may have to go online. Always get a sample before buying a huge stack of paper, and do a test print, to make sure that whoever or where ever you're printing from can actually use the paper you've selected with your design and medium. Here are a few other places to buy paper from.