Cutting External Threads
Thanks to the wonderful staff at the Urban Workshop, I have learned how to cut external threads on the lathe! This can be tricky because, in order to cut a particular thread-per-inch setting, you need to match the speed of the lead screw (which is moving the tool post along the z-axis of the part) to the rotational speed of the chuck. When they are matched correctly, your lathe tool cutting edge will trace the correct helix along the outer surface of the part for the threads-per-inch that you want. Let's look at this in detail:
Determine your threads-per-inch (TPI) requirement. For example, if you want to cut 1/4-20 threads, then you need 20 TPI. Side note: you should already have turned your part down to the major diameter of the threads you want (in this case, you would want to have your part turned down to 1/4" in diameter).
Put a neutral angle lathe tool in the tool post and align it to be perpendicular to the z-axis of the part - use the chuck face to align the tool post. Make sure the cutting edge is sharp and clean.
Check the table pictured below to adjust the lead screw speed to match the TPI you need. Following this table (thanks to Dave Markowitz for the image), for example, you would need to adjust gears A and D in the stud gear box. Every lathe has a slightly different collection of settings, but the table will tell you how to adjust each knob to make the lead screw move at the correct rate.
While you're adjusting the settings at the head stock, you should also slow down the RPM of the chuck. Thread cutting is tricky and you should spin the chuck as slowly as you can. Adjusting the RPM of the chuck will not affect the settings for the lead screw that you set in the previous step - the carriage will still move relative to the part at the correct speed to cut the desired TPI.
There should be a smaller table on the carriage itself near a rotating dial as shown in the image below. When you start the lathe, the dial will rotate - you will need to engage the carriage on the lead screw when particular intervals on the dial pass the dot in the bottom right hand quadrant. The table tells you which intervals should be used for a particular TPI - for example, for 20 TPI, you can use any interval, 1-8. When you start up the lathe, the dial will start to spin. Watch it until the desired interval mark passes the dot, then push the handle circled in the image below to engage the carriage with the lead screw. The carriage will then move at exactly the right speed to cut the threads on the part.
Before you start to cut the threads, I recommend you zero off the cross-slide against the part in the x-axis. I usually cut .010" - .020" at a time and do multiple passes to cut the threads. Dial in to your first cut diameter before starting the lathe. I would also highly recommend starting your cutting tool to the right of your part with a couple inches of buffer space. You will want to make sure you engaged the carriage at the right interval on the dial and that it is moving smoothly before you make contact with the part.
Repeat the cutting process several times, moving in on the x-axis each time. When the threads are well-established (.050" - .060" deep), you can chase them with a die to finish them cleanly.