Weight: Paper thickness is measured by weight. The higher number you see, in pounds, on the packaging label, the thicker your paper is. This doesn't mean you have to find the thickest paper you can—most card stocks are a great weight for invitations and can hold up to being mailed.
As a graphic designer and someone who pulled out all the stops designing her own wedding invitations, believe me when I say you can make DIY wedding invitations as simple or as complicated as you want. A lot of people go into making their own invitations hoping to save money, but I'll tell you up front: there are plenty of affordable places to get nice invites with less work than it will take you to DIY wedding invitations. However, if you have specific reasons to go DIY on your paper (like, you want to bring your specific idea to life, or you want to get creative) read on. (Fun fact: Every APW editor DIYed their wedding invitations, so we've got your back here.)