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.
ONLINE PRINTING: There's a pretty wide variety when it comes printing via online companies, so always make sure you can get a proof of your invitation. With some companies, that may cost extra, but it's usually not very much. Much like with at_home printing, your paper options will usually be limited to whatever the company carries, and each company's paper set is going to be slightly different. You can usually request paper samples for free. Here are a few companies I've used with good results: