The usual way to do this I think would be to use the correct bit in a router, but it could also be done using a suitable cutter chucked up in a drill.
If using a drill ideally it will be in a drill stand for accuracy and repeatability, or a drill press will be used.
Round-nosed router bit: [ Source ] 'Rotary file' or burr for drills: [ Source ]
It's a little unconventional but an alternative if using a drill or drill press is to use a ball-headed grinding bit.
I use one of these myself on very splinter-prone woods to form a countersink for screw heads: The above image shows the diamond-coated and abrasive stone versions.
These will grind wood approximately equally but in general you'll get better durability from the first type.
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I'd use a forstner drill bit sized smaller than the game balls in a drill press.
This will leave a flat bottom hole with a slight pilot hole in the center
but I think that would be OK for your intentions.
The drill press should have some sort of mechanism to control the depth.
If it doesn't, you could have your table low enough
so the stroke of the drill press completes at the appropriate hole depth.
I used this method to make a marble "game" for my children.
Each child starts with three marbles on a board, throughout the course of the day if they misbehave they lose a marble.
At the end of they day the need to have one marble to "pay" for dessert.
It's a lot of fun for me, I'm not sure they like it too much...
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There are various ball-end cutters available, but balls or ball-shapes also fit into cylindrical or conical holes (and cylindrical holes with flat or conical bottoms) more easily achieved with a "normal" twist drill bit, a center drill, or a countersink.
Depends what you want, or how much function is important .vs.
the way one or the other looks.
You will want a drill press (or a CNC Router) to maintain consistent depth (the router might also maintain spacing for you.)
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Perhaps not as cool as the other ideas you could always have this done in two pieces as well.
You could do this with two boards and a basic drill.
Top board
you could drill straight thru.
Use some sacrificial wood beneath to reduce tear out.
Then glue that to a normal board below to give the overall project some size.
Be sure to clean glue that would come out from the holes.
This could add a nice appeal if you use two different coloured woods.
Only obvious downside is that you would not have curved recesses with this method.
Single tool suggestion
A traditional approach for this would be to use something like a spoon bit.
Image from LeeValley Tools To prove my point the product description from the page.... One, it will bore a hole with a round bottom .
This allows maximum possible blind hole depth in a chair leg to receive a stretcher with no chance of a brad point or spur breaking through.
If you do plan on using these the minor con about this.
Working with them you need to remove the bit to periodically clear debris.
This should only be considered a minor hassle.
Also, since they have do not have a central spur or point, they can be hard to get properly positioned.
With frequent use its not so hard but making a small pilot hole certainly helps.
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One other option is a simple bowl gouge .
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I am making my own wooden board game. The game pieces have ball-shaped bases. I want them to move around the board and rest in small indentations. Something like this:    What kind of tools and techniques should I use?
Making a small indent/crevice/depression for a ball
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<technique><tools>
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S0: The usual way to do this I think would be to use the correct bit in a router, but it could also be done using a suitable cutter chucked up in a drill.  If using a drill ideally it will be in a drill stand for accuracy and repeatability, or a drill press will be used.  A forstner drill bit sized smaller than the game balls in a drill press would be another option.  There are various ball-end cutters available, but balls or ball-shapes also fit into cylindrical or conical holes (and cylindrical holes with flat or conical bottoms) more easily achieved with a "normal" twist drill bit, a center drill,  or a countersink. A traditional approach for this would be to use something like a spoon bit or a simple bowl gouge. A router and suitable bit is recommended. Otherwise, a number of bits are suitable, including a forstner bit, , ball-headed grinding bit or cutter, or spoon bit. A drill press is also advisable for accuracy. Another option is a bowl gouge.
Use the correct bit in a router or a cutter in a drill. A forstner bit is recommended. A drill press would give better accuracy. Drill press and ball-headed grinding bit. Spoon bit. Bowl gouge.