What does "Worst-case hardness" mean in lattice-based cryptography?
One line: worst means any and average means random.

Lattice-based cryptosystem

Let me restate. 
Fix security parameter $n$. 
What the reduction shows is the existence of a solver for the lattice problem on input any $n$-dimensional lattice using the adversary breaking a lattice-based cryptosystem with the security parameter $n$ on the average case.

Since we can solve any instance, we can solve the hardest one of dimension $n$.

ElGamal encryption - average-case

We have a reduction that converts a random instance of the CDH/DDH problem into a random problem of one-wayness/indistinguishability game. This means that the security is based on the average-case hardness of the problem.

ElGamal encryption - worst-case

Correctly speaking, the security of the El Gamal encryption scheme can be based on the worst-case hardness of the DL-like problems over the group instead of the security parameter.

Fix a prime-order group. We can convert any instance of the DL/CDH/DDH problem on the group to a random instance of the problem on the same group.
( This property is as known as the random self reducibility.)
Combining it with the previous reduction, we can say the security is based on the worst-case hardness of the problem over the fixed group.