I use scanning tunneling microscopy (STM), which is a similar probe technique to AFM. (It has the "restriction" that you can only measure conductive samples, whereas AFM can measure insulators, but it allows you to get additional electronic information in addition to topology.)
Our tips are made of polycrystalline tungsten wire, 0.25mm in diameter. We prepare them by a fairly standard electrochemical etching method, as follows.
1. A small beaker is filled with a solution of potassium hydroxide.
2. A ring shaped wire is positioned at the surface of the solution
3. The tip wire is suspended partially in the solution in the middle of the ring, and a voltage is applied between ring and tip wire.
4. This causes the tip wire to etch preferentially at the surface of the solution.
5. The weight of the section of the tip in the solution causes the area being etched to stretch out as it gets etched.
6. At some point, the lower section of the wire drops off. This has an associated drop in current, which we use as an indicator to stop the etching. (This is pretty important, if you don't stop the etching as soon as possible after dropoff, the newly formed tip gets blunter; the electronic trigger which detects current drop stops the voltage in about a microsecond)
7. The tip is rinsed in deionized water (to remove KOH residue) and inserted into ultra high vacuum ASAP.
By this method we routinely achieve tips sharp enough for atomic resolution measurements.
For standard AFM, I know that people typically use silicon nitride tips. I believe these are made by standard photolithography processes with a silicon wafer, and then the nitride layer is grown on top.
Early on, they were made by smashing diamonds with hammers and then looking for good candidates with an optical microscope. They don't need to be smaller than the atoms they are imaging. If you use a rounded 25mm broomstick end, you can easily feel a 1mm bump.
I don't know much about how they're made, though I'd imagine it's using the same techniques they use for all MEMs, which is basically what they are. You can read some basics on MEM fabrication process flow here:
But they're definitely not smaller than the atoms they're manipulating. The tips are in the nanometers to 10s of nanometers range in size I believe. Which is about ~100-1,000 atoms wide.