So you don't have a wood lathe? But you have a drill press. Well that's good news. Now all you need is few bucks, some basic handyman skills and about 45 minutes of your time.
This is an easy project that will make possible you having so much fun when you'll start lathing. Wood lathing is such a fun, it makes you feel like an artist. So let me show how easy you can make this:

WARNING: This is a homemade wood lathe for occasional use and not for everyday use and also for turning small pieces of wood. If you want to build something solid for turning larger and longer pieces of wood you'll need to use bigger materials like washing machine motor, bigger bearings etc.

For your safety wear eye/face protection and gloves

Materials you need:

A piece of hardwood preferably about 12" x 6" and about 1 1/2 thick
1x threaded M8-M12 bolt, 1x nut and 1x wood 'three prong tee nut' of the same size
A bearing preferably extracted from a sliding door bearing wheel
A washer and two screws to hold the wheel bearing in the place/hole
A long thick bolt to make the toolrest (at least 6" long and thicker it's better)

Tools you'll need:

Drill Press
Forstner bits
Measuring tape or caliper micrometer
Hack saw
File
Screwdriver
Desk Grinder
Two clamps
Drill bits

Step 1: Make the Spindle (the Bottom Part)

The sliding door bearing wheel I found had two coupled wheels, so I had to detach on of the couple to get one wheel together with the shaft. For this I used a metal drill bit to drill one side of the shaft in order to release the wheels from each other. Of course I took the wheel which has the shaft still in it and I grind (in the desk grinder) the end of the shaft to make it sharp. After that I measured the diameter of the wheel and chose the forstner bit of a same diameter and made a hole somewhere in the middle of the piece of wood. I then inserted the wheel in the hole and followed by capping it with a washer of a same diameter to. Then I screwed two screws in the two opposite sides of the washer to keep it in place together with the wheel. REMEMBER: after you drill the hole the same size of the wheel, you MUST drill with a forstner bit (the size of the inner part of the bearing) about 1/8 profound at least, to avoid the friction of the shaft with the wood.
After this all you have to do is to drill a hole for the "toolrest" bolt and screw it in. Refer to photos for an aproximate distance of toolrest and spindle. With this done you have finished the spindle.

Step 2: Make the Drive Center (the Upper Part)

Using the hacksaw cut the head of the bolt. Put it in a vise and with the file make four points. Take the wood drive-in nut and ply or cut the pointed pins. Insert first the nut and then the wood drive nut in the bolt. Insert the drive-in nut in the chuck and tighten it. You can still adjust the length of the bolt and when you'll have the piece of wood in place and tightened than you can tighten the nut also.

Step 3: Align Them

To align the spindle and the quill I firstly put a drill bit in the chuck and align it togethe with the spindle and then I replace the drill bit with the quill. Make sure you tighten the base of the spindle with clamps.

Now you're ready to go and give a try to your homemade wood lathe. Put a piece of wood and tighten it by adjusting the bolt (the quill) and after this tighten the nut.

Step 4: Enjoy

When you start lathing remember not to apply too much force with your chisel and always keep the chisel backed on the right side of toolrest. Have fun and if you like this project don't forget to vote it if it's in a contest.