The exact puzzle behind RSA: undo a "raise to a power" step without knowing the secret primes. Anyone who can factor can solve it.
The idea: RSA scrambles a number by raising it to a public power e and keeping the remainder after dividing by N. Toy example: with N = 33 (3 × 11) and e = 3, the message 4 becomes 4³ = 64, which leaves 31. Undoing that means taking a "cube root" in remainder arithmetic, which nobody knows how to do fast for big N. But knowing 3 and 11 lets you compute a secret power d = 7 that undoes it: 31⁷ leaves 4 again. Encryption: anyone can apply e, only you can apply d. Signatures run it backwards: only you can make a number that turns into the document's hash when anyone applies e.
Why it matters: If it falls, RSA encryption and signatures fall with it.
On the map: e-th roots mod N
Given a modulus N, an exponent e and a number y, find x with x^e = y (mod N), without knowing the factors of N. Knowing the factors (the trapdoor) makes this easy. That is how RSA decryption and signing work[1].
If you can factor, you can solve the RSA problem. The reverse isn't known, so it could in principle be easier than factoring, which makes it a slightly stronger assumption[2].
If any of these breaks, this breaks too.
If this breaks, these break with it.