Did you ever have a grandmother or great grandmother who would very carefully open her presents and save every scrap  of wrapping paper?  That level of conservation may be a little overboard for this day and age, but it is practical and enviromentally friendly to save and reuse some elements of packaging.

Gift bags are wonderful for reuse.  My best friend and I have a large one that we pass back and forth between us.  I wrapped several presents for my family this year in gift bags (and Christmas gift pails) that I received from my kids at weekday school.

Bows are also good to save for reuse.  As is the ribbon from larger packages (if it's over a foot long).

Wrapping paper has a tendency to get wrinkled and torn after use, so I rarely save it, unless it's a very large and undamaged piece.
(We do keep a carboard role however for unused wrapping paper scraps, pieces that aren't big enough for large gifts but could be used for smaller ones.)

Tissue paper on the other hand is usually good to save.  You might want a crisp, fresh sheet to look nice peeking out of a gift bag, but the crinkled tissue paper can be reused in for colorful stuffing around boxed gifts or for gifts that you're shipping.

All these things will wear out eventually.  After a bow looks crushed and the gift bag starts getting faded or wrinkly, it's time to retire them, but unless you're really rough with your packaging, those things should be good for multiple uses.

Bows and bags can cost anywhere from fifty cents to several dollars, particularly the fancier ones.  So cutting that cost out by recycling can ease your own wallet during a straining time of year.  Plus, less trash is better for the earth. :)

If you're running out of wrapping paper or would like to save a few pennies, you can wrap gifts in newspaper (I like using the funnies section for this, since it's colorful and entertaining...but if you're gift is for a sports nut, the sports section might be more appropriate).  And don't dismiss plain brown paper or butcher paper.  A "brown paper packages tied up in string" theme can be a very cute.  And if memory serves,  butcher paper is nice and cheap and can be bought by the foot or yard at some art supply stores in a variety of colors.