In our instrumental analysis class, our lecturer told that using peak height is totally wrong. 
  Never believe an absolute statement! 
 
 
I am little bit confused now. If we thinkof the peaks as triangles, a triangle may have the same area with different heights.
The triaangle 1 will be the poorly resolved peak(lower height) and triangle 2 will be the well resolved peak.(higher height)
So far so good.
  Actually, not so good. 
First of all, resolution, by definition, applies to two peaks, not one, so that "poorly resolved" does not necessarily imply "wider". To a first approximation, peak width for a given compound under given separation conditions is constant and independent of the amount injected (lots of assumptions here, including not overloading, but a safe general statement). That means that peak height is, in fact, approximately proportional to the area.
but if the resolution is same, then the height and the area will give the same result, won't it?
  Usually, yes.
so why using peak height is totally wrong.
 As suggested by other posters, it's not 
totally wrong. It is, however, usually considered to be unwise. Re-read my earlier statement about peak width being constant, and note that I qualified it ("lots of assumptions . . ."). If you overload by injecting too large a sample, or if your column gradually deteriorates so that you lose efficiency (i.e., get wider peaks), or if your mobile phase pH drifts so that the peak begins to develop a "tail", or . . . and on and on, then peak height can change while area stays consistent. That's why area is generally preferred.
As DR and I pointed out, there 
are situations where peak height is preferred, but these are the exception rather than the rule.
I will also discuss this point with my lecturer next lesson but I want to learn the right answer. He is telling a little bit wrong I think. 
 
 
  Definitely discuss it with him (but be diplomatic!). I'm not sure about "wrong"; perhaps "oversimplified" would be more accurate.