How do you drill a smooth round pit in wood?

You can get that with a drill bit.Look closely at the end of the bit and you will see.Pick one with a shallow profile and you will have your mini bowls/divots

Put a piece of tape on the bit at the depth you want. Then stop when the tape hits the wood. If the wood is soft you can tap a ball bearing into the hole to smooth it out. Or use a Dremel tool – but you might not have one handy.

Or use a countersink/deburrer had to look up the English word, don’t know if it is the right word?).
They look like this.

Personally, I would use a ball-nose end mill. I am a machinist, and we have hundreds of them lying around in all different sizes.

Picture

Go to mom and pop hardware store. Explain what you want to do. Buy drill bit they tell you to.

I second @worriedguy‘s idea. It’s the cheapest and easiest. Just wrap tape around the bit at the depth you want the hole, at a ¼ inch.

A Dremel tool would work well for what you’re trying to do. They make ton’s of different bits. Know anyone that has one you could borrow?

If the conical shape that a regular drill bit leaves is unacceptable and you really must have a hemispherical depression, one solution would be to regrind the tip of a drill bit so that it is dome-shaped instead of conical. This can be done on a simple bench grinder (in fact, I just did it to make sure it works). Round off the point at the tip, then round off the corners of each of the flutes.

It will have a harder time digging into the wood at the start of drilling, and will tend to skate out of position until the hole gets started, but that’s easily remedied by starting the hole with a smaller bit. The reground bit leaves a nice clean bowl-shaped dimple.

Perhaps what you want is a Forstner bit.

Maybe you want a ball end router bit?:
http://www.amanatool.com/bits-fv/45960.html

Use a countersink drill bit.

If bowl shaped concave indentation is what you want, I can’t see it happening with a drill bit of any sort or a router. You need a Foredom or Dremel, rotary grinding tool with a ball nose burr. Then you have to draw a circle on the wood and hollow it out by hand. If you want it perfectly symmetrical a lathe is the best bet. If you don’t have any of these tools, a curved chisel would be my next choice. @FluffyChicken has a good idea if they don’t have to be too large. What are you making???

@rooeytoo If you look, they make all sorts of bits (including ball-nose) for drills and routers. I just prefer to do that sort of stuff on an OKK CNC mill with a 42” table, if for no reason other than we already have a lot of ball-nosed end mills so I would rather not spend the money to get something that will do the same thing as something I already have access to. There is also the fact that is I need a bigger hole than I have a bit to do with a G81 (the FANUC command for drilling), I can just program it to mill out a pocket of arbitrary size/shape.
Of course, since the OP is talking about wood instead of steel, and only going ¼” deep, they could do the same stuff with a router or drill.

@jerv, I never have a clue what you are talking about, but there is no doubt in my mind you are right on track, heheheh. I have a Foredom in the shed and I love it so that is what I would use, course I never make anything that has to be precise, I like the hand made look. I do have a router and a tray full of bits for it, but nothing that would do this job. Next time I go to the hardware store, I will have to check out the assort router accessories. Cheers mate.

The CNC mill at my work is set up for graphite composite materials. The machinists would skin me alive if they found sawdust in their machine.

@RocketGuy That is why you clean up after yourself ;)