This quick origami envelope is a great model to learn. You don't need any glue because the folding itself will keep the envelope secure, although you can use a bit of glue if you wish. Any time you don't have an envelope handy, you can make one with this technique.

You can use any kind of paper to make an envelope. If you need an envelope for a birthday card, it can be the same wrapping paper you use to wrap the gift. Also think about using any kind of colored, patterned, or scrapbooking paper. Even simple printer paper or a brown paper bag will do.