Which ATC instructions should be repeated back by the pilot?
ATC is, broadly speaking, required to ensure that pilots hear, understand, and acknowledge information that is delivered.  The way they do that is via read-backs, and although the AIM mentions a few of the things that are required it's a good idea to read back anything that's important (If you're not sure, err on the side of "It's important"), with your aircraft callsign (No callsign? Doesn't count!). 

What are some important things controllers usually expect you to read back?


Clearances (headings, vectors, altitudes, transponder codes, and radio frequencies).
Like the AIM says, mutual verification.
Runway assignments, hold short instructions, line-up-and-wait, and takeoff clearances.  
Taxi routes Instructions, ESPECIALLY runway crossings &amp; hold-short instructions.  
Wake Turbulence advisories.
Traffic point-outs.
The magic words here, by the by, are "Negative Contact" and "Traffic In Sight".
I think if you say "Tally Ho!" you're required to buy everyone on the frequency a bottle of 30-year-old scotch.
Instructions to "Maintain visual separation with" some traffic
ATC is handing off collision avoidance responsibility to you as the pilot.
They need to hear "Will maintain visual separation with" the traffic in question.
They also need you to tell them if you lose visual contact.
Airport Point-Outs (Especially if there are several airports close to each other)
You can also pre-empt this by calling the controller: "Cessna 12345 has Podunk Municipal in sight", possibly with a bearing and approximate distance.
Landing Clearances
If you're in a busy pattern and they give you a sequence ("Number 3, cleared to land") they want to hear that part too.
That altimeter setting you get after a handoff.
(I blame Don Brown for getting me into this habit, but it IS a good one unless the frequency is crazy)


This list is by no means exhaustive, but it's most of the big stuff.