What is the difference between ∃x∀y and ∀y∃x?

Question

What is the exact difference between ∃x∀y(condition) and ∀y∃x(condition)? Translating these into English,

• ∃x∀y(condition) = There is an x for all y such that (condition) is satisfied.
• ∀y∃x(condition) = For all y, there is an x such that (condition) is satisfied.

These two seem to be different on the surface, but I could not grasp a vivid understanding of it. Also, I would want to ask how the difference in orders of quantifiers impact the final result.

Answer 1

The first condition is stronger then the second one.

An illustration: every person has a mother (his own mother), but there's no a mother of every person.

Answer 2

You have translated the sentence $\exists x \forall y$ (condition) incorrectly.

It means: there exists an $x$ such that for all $y$ (condition). This means that there is just this on single $x$ that works for all $y$ simultaneously.

The sentence $\forall y \exists x$ (condition), means that I can choose a different $x$ for every $y$.