Making wooden bowls is a woodworking skill that takes a day to learn but a lifetime to master. Most woodworking projects require many specialized tools that are expensive and for most people, take a lifetime to acquire. Turning wood on the other hand, requires only chisels and a lathe. You can make banisters, finials and baseball bats, but the real fun in owning a lathe is making bowls. You can spend years making different types of bowls and never get bored. I consider working with wood simply another medium for the artist but no where is this more true than with woodturning because it is completely free form, like sculpting.
One of my favorite things about turning bowls, is that you can use wood from trees in your area rather than heading to the lumber yard for wood. The lumber yard is to trees as the grocer's meat counter is to a cow; you get the end result but you miss the intimate connection to it's source. When you carve a bowl from a tree, you have a direct connection to the wood, the life of the tree and it's sacrifice to provide you with it's flesh. Making a bowl where you know the provenance of the wood will make the result even more special.
I'm making this bowl for a friend of mine and so I took a log from a beech tree that fell down in their yard. Every growth ring you can see on the bowl is another year of life shared with the tree. The bowl's thicker rings are warmer or wetter years that were especially good for growing or perhaps another tree nearby fell down, letting in more light. The narrow rings show years when there was drought or perhaps a late season ice storm that stunted that year's new buds.
When making bowls, it's imperative to have dry wood. Newly cut wood is green, meaning that the tree is still alive and there is still a relatively high moisture content. As you're not kiln drying the wood, like a lumber company would, you need to let the wood air dry for several months to a year in a low moisture environment. If you do not, the wood will split as the moisture dissipates. It would be a shame to put so much work into the bowl only to have it split. This log here has been aging in the woodpile for about two years and in a garage for about 4 months.
One note of caution, operating a lathe can be dangerous. It's best to learn how to turn wood with a simple project with someone who knows what they're doing. Like Norm says, be sure to read and understand how to use your power tools. And there is no more important tool than safety glasses.

A chain saw can help with the rough cut of the bowl. The round section of the bowl must be perpendicular the round part of the log. In other words, the center ring of the wood must go through the sides of the bowl, you cannot have the center ring of the wood in the middle of the bottom of the bowl. The way the grain runs, you'll end up with a warped or cracked bowl. Many people make bowls out of the root ball of the tree precisely for this reason; the grain runs in various different directions making the bowl stronger and less likely to split along the grain. These pieces of wood have the added benefit of being very beautiful.

First, remove the sides of the log.

The two flat sections will make up the top and bottom of the bowl, or in other words the base and the top rim. Take a compass or trace something circular to make the rough outline of the bowl.

Use the chain saw to remove the excess. You'll now have a roughly round bowl blank. The grain of the wood will run from one side to the other, not from the top to the bottom. You'll also have a lot of saw dust and scrap pieces of wood for the kindling pile! I believe Michelangelo said something to the effect of that the sculpture was always inside the marble, he just had to remove the rock around it. You'll find this true as you begin removing the wood around the bowl.

There are several ways to attach wood to a lathe. When making newel posts or baseball bats, you rotate the wood between two points, one that is connected to the motor and another that spins free. You start like this when making a bowl but once the piece is rounded, you'll need to spin the wood with only one connection to the router- the motor. This requires a much stronger connection.
There are a few ways to make this connection. One is to use an expanding chuck that fit insude a grove that you carve in the wood. This is done by screwing a plate to the wood, turning the wood until it is round, and then carving a grove on the bottom of the bowl. I prefer to skip this step and just screw the bowl to the mounting plate. I'll cut off the base of the bowl that I'm screwing into, so you'll never see the screw holes.

At this point the wood blank will be circular but not smooth. The first step is to put the lathe on it's slowest setting and remove material until the blank is smooth and round. If you spin the wood too quickly while it's still lopsided, the piece will vibrate uncontrollably and you wont get much carving done. The cupped chisel, called a gouge, is the best tool for removing the excess material. As the wood turns, the chisel will remove the high points of the wood until the entire surface is round.

When you remove enough material that the piece is round, you can increase the speed of the lathe. The faster the lathe spins, the smoother the cut the chisel will make.

With the piece rounded, you can begin shaping the wood. This is where you can get creative with the shape of the bowl. Let the natural shape of the wood guide you. Follow the shape and contours of the wood. When you're finished removing material and want to focus on shaping and smoothing, move on to the other chisels.

Removing wood from the middle of the bowl can be difficult because it requires cutting a fast moving narrow inside edge. Making wide open bowls is much easier than making narrow bowls for this reason. With some practice, and moving of the guide fence, you can make narrow walled bowls like this one. However, chisels can grab and cause injury easily, so be very careful.

Eventually the center will be hollowed out and you can begin shaping the bowl in earnest.

When shaping the bowl, using a caliper is a useful because it allows you to keep the thickness consistent across different sections of the bowl.

Different chisels have different purposes. I still have a lot to learn about bowl making, but generally this flat chisel is used for smoothing and shaping.

When the bowl is the shape you want, begin sanding while still on the lathe. It is much easier to do it now than when it's removed from the lathe.

When removed from the lathe, unscrew the mounting plate...

... and remove the hunk of wood that the mounting plate was screwed into. How you remove this will depend on the tools in your workshop, but a handsaw will do the job nicely. Then just sand the bottom flush.

Some hand sanding is always a good idea.

Finishing the bowl off with some steel wool will add polish and shine.

I've heard several different schools of thought on sealing bowls. I prefer not to use mineral spirits simply because they're petroleum based however I've heard some woodworkers recommend them to seal bowls and cutting boards. One thing for sure, do not use olive or vegitable oil as they will eventually go rancid. It will not hurt you, but they may have an unpleasant smell. Nut oils are the best choice as far as I am concerned. They're all natural and will not go rancid. I've heard that walnut oil is the most stable nut oil for sealing wooden bowls.

The oil also nourishes the wood, prevents cracks and splitting and brings out the beauty of the grain.

The heartwood runs directly through this bowl and there is even a hole in the center growth ring. The pigment of the heartwood is much darker than the sap wood further out. It almost looks like a paint spatter on the edges. Very beautiful.

And when you're proud of something that you've made, be sure to sign it with your initials. Buying a branding iron is a great investment, but you can always carve your initials in a hidden spot.