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.
Letterpress: The designer and paper lover's dream, letterpress is visually lovely and highly coveted all over Pinterest. However, it is expensive because it is a fine art that requires some heavy machinery. Letterpress is the option to use when you want to support the artists making it, not because you think you should. Thomas Printers is a great option for people looking for letterpress invites on a reasonable budget. The other option to letterpress is to build your own, but you better have a serious love for the actual craft. The majority of your guests, sadly, won't know or won't care that your invites are letterpressed.