From g^a and g^b, compute g^ab. That's exactly what an eavesdropper on a Diffie-Hellman exchange would have to do.
The idea: In Diffie-Hellman, Alice publishes g^a and Bob publishes g^b. Each raises the other's number to their own secret power, and both reach (g^b)^a = (g^a)^b = g^(ab). An eavesdropper sees g^a and g^b but has neither a nor b. CDH is the bet that combining two public halves into the shared secret is impossible without one of the private exponents.
Why it matters: Its hardness is what keeps classic Diffie-Hellman keys secret.
The details
On the map: in Fp*
Given g^a and g^b, compute g^(ab). If discrete log is easy, so is this: find a, then compute. Whether the two are equivalent in general is open[1].
CDH being hard is what makes classic Diffie-Hellman key exchange secure (KE)[2].
This assumption gives key exchange on its own, so it needs more than one-way functions and falls if public-key encryption is impossible.
TD
This assumption gives trapdoor functions on its own (computable by anyone, invertible only with a secret), so it falls if trapdoor functions don't exist. Schemes built on it need not use a trapdoor: ElGamal doesn't.
References
Ueli M. Maurer and Stefan Wolf. The Relationship Between Breaking the Diffie--Hellman Protocol and Computing Discrete Logarithms. SIAM Journal on Computing, 1999. doi:10.1137/s0097539796302749
Whitfield Diffie and Martin E. Hellman. New Directions in Cryptography. Democratizing Cryptography, 1976. doi:10.1145/3549993.3550007