Your public key is your private key "multiplied" by a fixed point on a curve. Going back from the public key to the private key is believed to be infeasible.
The idea: Your private key is a random number x, and your public key is the point you reach by "adding" a fixed curve point G to itself x times, which is fast thanks to repeated doubling. Going back from the public key to x is the hard direction: the best known method needs about 2^128 steps. A signature proves you know x without revealing it, which is all an account needs.
Why it matters: This single puzzle protects every Ethereum account's signatures.
Adding points on a curve
The curve: the addition rule. Draw the line through P and Q. It meets the curve at one more point. Flip it over the horizontal axis to get P + Q. Adding a point to itself works the same way, with the tangent line. The dots: the same rule with whole numbers, counting modulo 67. Ethereum's secp256k1 uses this same equation, but modulo a 78-digit prime instead of 67. Press play: each jump adds G once more, and the landing spots scatter with no visible pattern, so seeing where you landed tells you nothing about how many jumps it took. Here you could check all 78 possibilities. On the real curve there are about 1077.
The details
On the map: best attack: Pollard rho (~128-bit)
Given a public key Q = x·G on an elliptic curve, find the private key x. For secp256k1 the best classical attack, Pollard's rho, needs about 2^128 operations[1, 2], far beyond any computer.
J. M. Pollard. Monte Carlo Methods for Index Computation (mod p). Mathematics of Computation, 1978. doi:10.2307/2006496
Standards for Efficient Cryptography Group. SEC 2: Recommended Elliptic Curve Domain Parameters. Version 2.0, 2010. secg.org/sec2-v2.pdf
Peter W. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing, 1997. doi:10.1137/S0097539795293172