How many ways can you rotate / flip the triangle so it looks the same before & after?
Taken from this video
What she did is defining $6$ basic transformations. That is,
$$\{1, r, r^2, f, r\cdot f, r^2\cdot f\}$$
We need to find multiplications that results $1$.
Now, this is really a modular arithmetic so for example $r \equiv r^4$, so we need to count what's known as congruence (?)
We know that $r^3 \equiv 1$ and $f^2 \equiv 1$.
Now we only need to find the number of permutations and that is:
$${{3} \choose {1}} \cdot {{2} \choose {1}} = 6$$
Is that correct? Can you elaborate more about congruence in this context?
First, you ask about how to find the number of ways.
To find the number of transformations of some shape it is best to simply consider the vertices and their possible positions. For the equilateral triangle, the first vertex can be fixed in one of 3 positions and then another vertex has 2 possible positions. This completely fixes the triangle and so the number of transformations is 3 times 2 i.e. 6.
You also enquire about the relevance of congruences.
When dealing with repetitions of one particular transformations, e.g. five rotations of a 120 degree rotation, it is indeed useful to see the connection with congruences. In this example we have 5 such rotations being the same as 2 such rotations.
Groups which only involve repetitions of one particular transformation are called cyclic groups and when you study these you will find that your knowledge of congruences is very useful.
However, from the video you will have seen that this is only part of the story for general combinations in groups.