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.
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: