So you're not satisfied with a skinny rope, a granny-knot, and a branch that wouldn't hold a squirrel? Learn to do it the right way.

An eye bolt will keep your rope from rubbing the bark off of the tree and/or wearing through. It'll also make the swing much smoother.

A monkey's fist will give you a place to sit (or stand) while your swing. And, let's face it, it will look awesome.

Step 1: Step 1: Get Your Hardware

For about $15 at Home Depot or Lowes, you can get:

1x clevis shackle (a nice big one; mine was rated for something like 2000 lb)
1x eye bolt (long enough to go through the branch you've chosen)
1x regular nut
1x lock nut
2x washers

If you don't already have a rope, get one while you're at the hardware store.

My rope is nylon, about 30 feet long and about 1.5 inches thick. It cost me somewhere around $20.
You need 10-15 feet of rope for the monkey's fist knot
Thicker is better: no one wants to cut their hands on a thin rope, even if it's strong enough.

Step 2: Tie the Monkey's Fist

Do this first.

You can't judge exactly how long the rope will be once this knot is tied. If the rope is already hung, you'll probably end up with a knot that is too high or dragging the ground.

The concept is pretty simple. This link shows all you really need to know.

The execution is more difficult, especially with a big rope. The images above show my method.

Step 3: Get Ready to Attach to the Tree

Assemble your hardware:

washers and nuts on the eye bolt threads
clevis shackle on the eye of the eye bolt
rope looped through clevis shackle and tied in a knot (an overhand loop is easy and strong)

If you're working alone (like me), put tools in a bag that you can hoist up into the tree.

Step 4: Attach the Rope

Climb the tree (or use a ladder, if you have one)
Drill a hole in the branch (yes, this is better for the tree than having a rope wear away it's bark)
Put your eye bolt through and tighten the regular nut over the washers
REALLY tighten the lock nut against the regular nut

Step 5: Enjoy

You now have a safe swing that should last you about a bazillion years (if not more).

Step 6: A Note on Tree Health

I'm not an expert on trees, but I've done some research on the best way to hang a rope swing without damaging the tree, and this is what I've concluded:

Protecting the tree's bark is important. If a tree or branch is girdled, any tissue above the damaged bark will die.
Looping a rope over a tree branch is likely to damage the bark. This is mentioned in many of the articles I found, and I witnessed it happen in my first attempt at a rope swing (see photo above).
Carpet or a garden hose can be used to reduce the friction of a rope loop, but it can be tricky to absolutely eliminate the friction and avoid damaging the bark as the tree grows.
A better solution is to use an eye bolt. This is the technique used by arborists to support branches, the technique used by this famous tree-swing hanger, and the technique recommended by almost every result of a google search for "rope swing and tree health". A small, clean hole is unlikely to damage the tree, and according to several sources the tree will tend to grow over the bolt completely after a few years.