The same runs of numbers appear over and over, so knowing the answer makes you faster. But these names are not single moves — 1-2-1 takes three, 1-2-2-1 takes four, and the order is written under each one.
1-1
The 1 at the wall sees two squares, and the 1 beside it sees both. Both point at the same single mine, so the square only the second 1 can see is safe. Open it and the rest falls out by counting.
3 moves — safe by comparing → mine by counting → safe by counting
1-2-1
The squares under the two 1s are mines and the one under the 2 is safe. This is the run people memorise first, and it keeps appearing as you work along a wall.
3 moves — mine by comparing → safe by counting → mine by counting
1-2-2-1
The squares under the two 2s are mines and the ones under the 1s are safe. It is easy to mix up with 1-2-1 because the answer is inverted — the mines sit under the larger numbers.
4 moves — mine by comparing → safe by counting → mine by counting → safe by counting
Why they work
All three come from one thing. When everything the first number sees is also seen by the second, the squares only the second one sees hold exactly the difference between them. A difference of zero means safe; a difference equal to the number of squares means all mines. The run is the answer you memorise — this sentence is the reason.