All you need to make a paper gift bag is one rectangular piece of paper, some scissors, double-sided tape, and a glue stick. Your inner three year old is going to love it.
The paper I am using for the tutorial is roughly A4/letter size, which makes a small but useful gift bag, and for an even smaller party favour bag I would halve it to A5. Don't get carried away with exact sizes though, as long as it's a rectangle, you're good.
Side note: you can use any kind of paper you like, but heavier paper does make for a better shaped end product.
Fold the short sides of the rectangle so that they meet in the middle, overlapping by a little bit more than the width of double sided tape. Run the tape down the length of the overlap and secure to other side.
Take the end of your paper that will form the base of your bag, and fold it up by a good 4cm (1 5/8 inch). Use a ruler or the handle end of your scissors to mark the creases sharply - this applies to all the folds and will help shape the bag nicely when you open it out.
Take the top edge of your fold and pull it down, while holding the underneath edge in place. This will form two triangles pointing away from each other.
Take one edge of your base and fold it back in, just over the centre line.
Open it back out and put a spot of glue on the small triangle that has formed on the outside edge as shown. What you are trying to do is glue the fold down, without gluing the base to the inside of the bag.
Fold and glue the other edge (add a touch of glue to the same small triangle but also this time on the outside edge so that it will stick to the fold underneath it, forming a nice solid base). As always, crease folds as sharply as possible.
Take the outside edge of your bag, and fold it in towards the middle, so that the bottom corners mitre, keeping the folds even all the way up the sides. Repeat on the other side.
Now gently open out your bag. It will form the basic shape of your finished bag, except for the side creases. This is where the difference between thin paper and heavier will show up - if you have used thin paper you will need to spend a bit of time fiddling about with the creases, especially at the base, to make them all sharp and pretty. It's no biggie.
Now invert the side fold as above, taking it all the way down to the base.
And there you go, one finished bag. To close the top I just fold it over and peg it down with a tag, but you could sew it shut, or staple it, or punch a hole and tie it with string. Etc., and so forth.