This is a brief summary about a wood-turning workshop taken at Prototyping Summer-school @ Howest IDC (Belgium).
http://www.howest.be/summerschool2012/

We did start with roughly shaped pieces of wood (quite cylindrical).
The heel was already present, and you'll need it in order to create the inside of the bowl.

For your own safety, make sure to have decent work clothing, no loose hair, dust mask and safety glasses.

EDIT:
In this guide I did mention a big and a small chisel.
The big chisel is a roughing gouge and the small chisel is a bowl gouge. (thanks pudtiny)

For safety concerns, it is best to create the round contour with a bowl gouge. Once the shape is approx. what you are looking for, then you can switch to roughing chisel. (thanks TimS124)

Don't push the sandpaper too hard, it might get burnt (and your fingers too). You can use a block of cork to hold the paper with (thanks Vidar_76)

Step 1: Creating a Cylindrical Shape.

Use the big chisel to create an flat outer contour.

Step 2: Outer Shape

With the big chisel you can make the rough line of the outer contour. A smaller chisel can be used to have a better surface so less sanding is needed.
Do not cut away the heel, as it will be needed to create the inside. You can make it more cylindrical with a small chisel.
The tool mount can be adjusted so the tool rests closely to the piece you are working on.
It should be at center height while cutting. The tool itself should be approx. 20° below the horizontal line.
The tip of the small chisel has to be pointed in the cutting direction.

Step 3: Sanding

Most important step of creating wood turned pieces it the sanding.
We did use 4 sanding steps.

Start with high grain size sand paper.
The sanding paper becomes hot, so you might use an old towel or gloves so you don't get burned.

Sanding paper:
P50
P120
P180
P400

Step 4: Heel

You can cut away a piece of the heel so the bowl won't wobble when standing.
Make sure to have enough strength in the heel so you can pick it up to finish the inside contour.
The small chisel is the best tool for this job.

Step 5: Finishing Outer Contour

Use an old towel to get most of the dust away.
Then you can put Cellulose sanding sealer onto the towel and apply it to the sanded surface.
The bowl changes color. Use a towel without sanding sealer and push it on the outer surface.
This removes the superfluous sealer.

Then you can add wax. Push it firmly against the outer contour.
The bowl will become more dark and the outer contour is finished.

Some of the holes inside the wood are woodworm holes. Can't do much about them, and they do make the bowl look even nicer.

Step 6: Preparing the Inner Contour

Place the bowl in the claws with the heel.
Drill a big hole in the center of the bowl. This can also be used as a reference for the depth you'll be drilling towards.

Step 7: Inner Contour

Same steps as the outer contour.
Don't cut deeper then the hole drilled in the center of the bowl!
You can start with a big chisel and end with the small one.

Step 8: Sanding

Again, this step is very important and needs to be done thoroughly.
Pay attention at the outer edge of the bowl, it needs to be sanded so it is round and smooth.

Step 9: Finishing Inner Contour

Same steps as the outer contour.
Start with a towel to remove most dust.
Use the sealer to seal the holes.
Then use a towel again to remove the superfluous sealer.
Apply wax to the inner contour.
Heat the wax with a towel until the color changes.