How does Diffie–Hellman differ from elliptic curve Diffie–Hellman?
You might want to checkout Wikipedia page of elliptic curves to get a basic overview.
The difference between DH and ECDH is mainly the group which is being chosen to compute the secret key(s). While DH uses a multiplicative group of integers modulo a prime $p$, ECDH uses a multiplicative group of points on an elliptic curve:

Alice and Bob agree on an elliptic curve $E$ over a Field $\mathbb{F}_q$ and a base point $G \in E/\mathbb{F}_q$.

Alice generates a (random) secret $k_A$ and computes $P_A = k_A P$.

Bob generates a (random) secret $k_B$ and computes $P_B = k_B P$.

Alice and Bob exchange $P_A$ and $P_B$.

Alice and Bob compute $P_{AB} = k_a P_B = k_b P_A$.
The secret $k_A$ and $k_B$ is a random value $\in \{1, ..., n-1\}$ where $n$ is the order of the group generated by $G$.


The equivalent RFC may be RFC 4753 (not sure, not into RFCs at all).
To agree on a common curve Alice and Bob may either generate a curve them selves or use predefined curves and base points. For this there are a couple of standards defined, see:

Brainpool
IEEE P1363
NIST
NSA Suite B
ANSI X9.63
...

Depending on application for ECDH you might want to choose different curves since some curves have benefits, for example amount of calculation steps for point multiplications.