Is energy relative or absolute? Does gravity break the law of energy conservation? Imagine a meteor, with a mass of 1 kg, traveling towards the earth at a velocity of 1 mile/hr. It is having very little energy, as it can easily be brought to rest. Now as it enters earth's gravitational field, its velocity increases. Now it has very high potential energy, given by mass$\times$gravity$\times$height. So how did it gain this energy? How is total energy conserved here? Is energy relative or absolute?
 A: The statement that gravitational potential energy is $U=mgh$, with the height $h$ measured relative to some arbitrary vertical zero, is an approximation.
The potential energy associated with the gravitational interaction between two masses $M$ and $m$ is given by
$$
U(r) = -G \frac{Mm}{r},
$$
where $G$ is an empirical constant and $r$ is the separation between the two masses.  The usual connection $\mathbf F = -\mathbf\nabla U$ between force and potential energy gives the usual inverse-squared force law.
This also has the nice feature that the interaction energy $U$ goes to zero if the distance between the two masses becomes very large.
Since only changes in potential energy are measurable (at least, in classical physics), having a negative gravitational potential energy everywhere is not a terrible flaw.
If you're near the surface of a planet with radius $R$, and your distance from the center of the planet changes by some height $h\ll R$, you can use the 
binomial approximation
\begin{align}
(1 + \epsilon)^n 
&= 1 + n\epsilon + \frac{ n (n-1)}{2!} \epsilon^2 + \cdots
\\&\approx 1 + n\epsilon
\end{align}
to find the change in the potential energy:
\begin{align}
U(R+h) &= -G\frac{Mm}{R+h}
\\ &= -G\frac{Mm}{R} \times \left(1+\frac hR\right)^{-1}
\\ &\approx -G\frac{Mm}{R} \times \left(1-\frac hR\right)
\\&= -\frac {GMm}{R} + 
m \left( \frac{GM}{R^2} \right) h
\\ U(R+h) &\approx U(R) + mgh
\end{align}
The approximation fails if your height changes by a substantial fraction of Earth's radius, which seems to be part of your confusion.
A: The meteor has potential energy while out in space. That gets converted to kinetic energy as it falls. When it hits the earth, most of that energy is spent on heat and deformation of the ground. The Earth will also move imperceptibly, having gained some of that kinetic energy.
By convention, we regard the potential energy of an object in empty space to be zero. As the meteor falls, its potential energy gets negative. But you could set the energy in empty space to be something else. There is no true reference, and the energy of something can depend on your reference point. From each of those reference points, however, energy will always be conserved.
EDIT: When we use the surface of the Earth as our reference, we often use the term $E_p = mgh$. That means we have chosen that an object at the surface has zero potential energy. All energy numbers change in that system, but changes over time in those numbers will be the same. The problem with reconciling the two systems is that $mgh$ assumes a constant gravitational pull $g$. That is no longer true when you are approaching space, because gravity gets weaker. So actually, $mgh$ is an approximation!
When you have an object coming from outer space, it's better to use the potential energy of a gravitational field.
$E_p = -G\frac{mM}{r}$
It assumes zero in outer space as opposed to on the ground, but that is really just a matter of perspective. A change in this for two different $r$ will be the energy that gets converted to kinetic (and ultimately heat/deformation) energy.
EDIT2: To answer your other question, I am not entirely sure about this. I would say the energy of the universe is absolute, but where that energy appears to be located depends on your choice of reference frame. Potential and kinetic energy are easily relative locally, but energy of rest mass and heat should be absolute even then. Don't take my word for it, though.
A: Ketil's answer covered well the total energy that includes the gravitational potential energy and kinetic energy. As he made very clear in those cases the zero point for energy, or for potential energy, is really arbitrary. Sometimes zero potential energy is at infinite r, other times you can pick the surface of the earth as the zero potential energy term. 
You question of whether it is absolute or relative, I.e., whether we can always treat it as relative to the zero point which we can pick, in all cases, is that no, you can't. That is because $E = mc^2$, so that energy is equivalent to mass, in some way. Well, in both Newtonian and General Relativistic gravitation mass creates a gravitational effect in the motion of bodies, which can be observed. m is not only a mass that can cause a gravitational effect on another's body (that m is called gravitational mass) but it also the inertial mass of an object (I.e., how much gravity can disturb it). 
So, non-zero energy in fact will create a gravitational effect, and will also be affected by gravity. Light, which is pure energy, is bent by gravity. It's been observed, early in the 20th century. Energy also creates gravity, but it has to be a large enough amount of energy so we can measure the effect. In most astrophysical situations that's just not big enough. For cosmological  (I.e., very large astrophysical distances such as 100s of millions of light years), the effect of the radiation (I.e., pure energy from mostly electromagnetic waves) does have an effect on the motion of the galaxies far away, which we can measure and use it to estimate the amount of radiation in the universe. 
We do have to use General Relativity's (GR) Einstein Field Equations  to calculate cosmological quantities. The measurements have all been fully consistent with those. It is using those that the Big Bang Theory was arrive during at, and the universe's expansion. There's a lot more to that beyond your questions. See the wiki article on it at for example, https://en.m.wikipedia.org/wiki/Lambda-CDM_model
It gets also pretty interesting for Black Holes colliding and merging, such as what was observed in 2015 and announced in early 2016 by the LIGO team. That also requires GR to calculate. The rest masses at infinity (or far enough away from each other) for the two BHs amounted to about 65 solar masses. As they got closer their potential energy (close enough calculated from the same Newtonian equations) became more and more negatives, and the kinetic energy increased equivalently. As they got closer they moved faster, and they they orbited around each other before joining they were moving at relativistic speeds, about half the speed of light as their BH horizons merged. The calculations as they were moving faster needed GR to calculate them right. They quickly merged and settled down to a single BH. The mass of the final BH was about 62 solar masses. 3 solar masses were radiate due away as gravitational radiation (which we detected 1.3 billion years later). 62 + 3 = 65, so total energy was conserved globally, if you include the energy of the gravitational waves. 
If those two BHs , including as they merged, were the only things in a nonexpanding universe the total global energy does get conserved. Also, the total gravitational effect on some other body (say we were a body far enough away) would be the gravitational effect of 65 solar masses, before they merged, and after they did and release all those gravitational waves, with their own energy. So, far enough away (to not affect their merging) we always would have been feeling the gravitational effect of 65 solar masses. Or really, until the gravitational waves passed us and went behind us, and then we'd feel the 62 leftover solar masses, plus some effect both in front and behind for those 3 solar masses, now partially ahead of us and partially behind. 
So, yes, energy gravitates. 
So, yes, absolute value is important. Well, the exception may be that there is no clarity's yet on how to treat energies due to nonzero fluctuations of quantum fields. Combining GR and quantum theory is still a work in progress (though we do know a few things, like that a BH can in fact loose mass to quantum radiation called Hawking radiation, but very slowly for big BHs). 
And so, yes, energy-mass gets conserved, but only in certain situations in GR, such as when spacetime is flat at infinity (which I assumed on top). It is stranger and not conserved, in general, for instance in our expanding universe. 
Hope this answers some of your remaining questions that were not treated in the first answer. 
A: In the two cases you use a different choice for you zero of potential energy. In one case it is at infinity, in the other at the surface of Earth. Also your potential expression is only valid near the surface of Earth. 
