First, we start with the symbolic statement:

!∀x∃y P(x,y)

Next, we replace the variables with English names:

!∀black thing ∃ raven-nature

Next, we replace the symbols with English phrases:

Not every black thing has raven-nature

Then we clean up the English:

Not every black thing is a raven.

We can repeat the process with the other sentence, being careful to use the same words when we replace the variables:

∀x∃y !P(x,y)

becomes

∀black thing ∃ not-raven-nature

becomes

All black things have not-raven-nature

and finally:

Every black thing is not a raven.

(I should note that my English interpretation of ∃y P(x,y) is probably a bit different and more compact than PhilGoetz's, but I think that's a linguistic rather than logical difference.)