In typical ECC-based cryposystems, the one-way function is based on the Discrete Logarithm problem as in classical Diffie-Hellman key exchange; only working in a different finite group, built with the help of some Elliptic Curve over a finite field, rather than in the group obtained directly by multiplication modulo a prime.
In classical Diffie-Hellman key exchange with parameters a large prime $p$ and generator some integer $g$, the private key is some integer $x$, and the public key is $y=g^x\bmod p$. This is equivalent to multiplying $g$ by itself $x$ times in the multiplicative group $\mathbb Z_p^*$; though actually, this is done with $\mathcal O(\log(x))$ such modular multiplications, using e.g. exponentiation according to the binary representation of $x$.
In ECC, we just use a different group, and typically note the group law $+$ and the generator $G$ (uppercase is used for members of the group, while lowercase denotes integers). The private key is some integer $x$, and the public key is $Y=x\times G$. This is equivalent to adding the generator $G$ to itself $x$ times per the group law; though actually, this is done using $\mathcal O(\log(x))$ such additions in the group, using techniques as above.
In both cases, obtaining $y$ or $Y$ from $x$ is easy, but obtaining $x$ from $y$ or $Y$ is believed intractable (for proper choice of group and generator, and $x$ random in $[0,n)$ where $n$ is the order of the generator).
Elliptic Curve Diffie-Hellman works essentially identically as its classical counterpart in $\mathbb Z_p^*$, except that the Elliptic Curve group allows shorter public key and more efficient calculation for the same security level. The same transposition from $\mathbb Z_p^*$ to Elliptic Curve group works for ElGamal encryption (which becomes the basis of ECIES) and various signature algorithms including Schnorr and DSA (becoming EC-Schnorr and ECDSA).
