I think that the separate parts of this problem are related enough to count as one problem, so I'll answer them all. Due to the flexibility of English, there are some variations possible with the wording here. Three people could give three slightly different answers, all of which are correct.
With that in mind, the if-then form is "If two angles are vertical then they have equal measures."
The hypothesis is what we are assuming. In this case, we're assuming that the angles are vertical, so that's the hypothesis.
The conclusion is what we say is true based on our hypothesis. Here we are saying that given our hypothesis, the angles have equal measure. That's the conclusion.
The converse is what you get when you switch the order of the hypothesis and conclusion in the if-then statement. If we do that here, we see that the converse is "If two angles have equal measures, then they are vertical angles." Note that while the original statement is true, the converse is false. This is often, but not always, the case.
The contrapositive is what you get when you negate the hypothesis and conclusion and switch their order. In this case, the contrapositive is "If two angles do not have equal measures, then they are not vertical angles."
The inverse is what you get when you negate the hypothesis and the conclusion but keep the order. In this case, the inverse is "If two angles are not vertical, then their measures are not equal."
I saved the inverse for last because unlike all the others, I honestly can't figure out what the point of it is (besides being a question to answer in a textbook). I'd be interested to hear from anyone who has ever had a reason to consider the inverse of a statement.