Vellum is a semi translucent paper similar to parchment. Most vellum sold for paper crafts is a plasticized cotton. It lends a softness to cards, adding layers and a delicate, feminine look.
Vellum is sold in a variety of colors as well as white. It is also sold with embedded patterns, pre-printed greetings, or as pre-cut or metal-rimmed tags.
In my opinion, the trickiest part of using vellum is adhering it to the cards. Because you can see through it, adhesives must be hidden with embellishments. You can also use brads instead of adhesives. There is a product called vellum tape, but I have never used it. (If you have experience using vellum tape, please leave a comment and let us know how it works.)
Caution should be used when heat embossing, because vellum burns more easily than regular paper, and can melt or warp if heated too long in one spot.
When stamping on vellum, heat set your ink, even if you use dye ink. It takes a long time for ink to dry on vellum because it doesn't soak into the paper; it just sits on top.
Special thanks to Julie for letting me use pictures of some of her cards (1st, 2nd and 4th). I'm also including a picture of one of the luminaries she made last year for Christmas presents. Little electric candles set inside shine through the vellum windows when lighted.