Predicate logic: What are the differences between ∀ and ∃ when it comes to comparing two variables?

Question

Say I have the four following logical statements, all over the domain of all integers.

1. (∀a,∀b)[a>b]
2. (∀a,∃b)[a>b]
3. (∃a,∀b)[a>b]
4. (∃a,∃b)[a>b]

I feel like they're all asking practically similar things, but I'm getting on confused on what exactly for all means. I'm writing what I think each statement is asserting literally, please correct me if I'm wrong:

1. All integers are greater than each other? (This is the one I'm struggling the most with)
2. There is an integer b that is less than all other integers
3. There is an integer a that is greater than all other integers
4. There is an integer a that is greater than an integer b

So if this is the case, I'm assuming that all are false except for (4). But in the event that I have correctly understood all four of these statements, how would you express something like how for all known integers, there is another integer that is greater and/or lesser than it?