How are absolute enthalpy and enthalpy change defined?
Enthalpy $H$ is defined as in your question. The most basic definition of enthalpy change when going from state 1 to state 2 is the enthalpy of state 2 minus the enthalpy of state 1:
$$\Delta H = H_2 - H_1$$
What is the clear difference between enthalpy and heat?
Enthalpy is a state function and heat is not. Heat is energy transferred from a hotter system to a cooler system (not to be confused with thermal energy, that part of internal energy due to the kinetic energy of particles randomly moving, i.e. what makes a sample have a certain temperature).
A simple experiment to experience the difference: If you rub your hands together, they warm up. If you wait a while until they are back at the original temperature, they will be in the same state (no change in enthalpy). At the same time, there was a heat exchange with the environment, increasing its temperature a bit.
In essence, some mechanical energy was converted into thermal energy. There was some heat transfer $q > 0$, but the enthalpy did not change ($\Delta H = 0$) in this thermodynamic cycle. So enthalpy change and heat are different concepts, and there are cases where their values are different.
Is it the constancy of pressure what makes the enthalpy a state function?
No, it is because of how enthalpy is defined. The internal energy is a function of state, and P and V are a description of the state. This means that by the definition of enthalpy, the same state will have the same enthalpy. (If you want to know why internal energy is a function of state: it is related to the first law of thermodynamics, but more fundamentally - it is what all measurements so far have shown).
There is a reason why looking at enthalpy instead of internal energy is often simpler, see below, related to how much of what we experience is at constant pressure (usually atmospheric pressure).
My teacher told me in very simple words that when heat goes inside a system, it is called enthalpy. Is this correct?
This is a shortcut for the following: If you run a process in a way that the only forms of energy transferred between the system and the surroundings are heat and pV-work, the change in enthalpy will be equal to the heat transferred. (In my example above, there was mechanical work; that would break the equality of $\Delta H$ and $q$, as would electrical work done by or on the system, or electromagnetic radiation going in or out.) Also, the system has to be closed (no matter entering or leaving the system).
pV-work is a term for the work of expanding the system's volume by $\Delta V$ into the surrounding. When this happens, you work against the pressure of the surrounding. Enthalpy is defined in a clever way so that it does not change due to pV-work at constant pressure:
$$ H = U + PV$$
$$ \Delta H = H_2 - H_1$$
$$ = U_2 + PV_2 - (U_1 + PV_1)$$
$$ = \Delta U + P\Delta V$$
It turns out the change in internal energy due to pV-work (at constant pressure) is $-P\Delta V$ (it takes work to push against the surrounding, done by the system, so the system loses energy), which cancels out against the $P\Delta V$ term in the calculation above. Thus there is no change in enthalpy due to pV-work at constant pressure.
Would constant pressure make the volume constant in a thermodynamic system?
No. Take an ideal gas (in a closed system) whose temperature rises for some reason. If you keep the pressure constant (by having the gas inside a floppy mylar baloon), the volume will increase. Enthalpy is a useful state function for systems at constant pressure because it does not change just because the volume changes.
Why is this answer so long and complicated (I added this question)?
At my college, enthalpy is introduced as a "fact" in our first chemistry course. A semester later, we learn about first and second law of thermodynamics, and we discuss the tools of thermodynamics a year or two later. So it is not taught from first principles, and you have to learn some stuff without a full picture of the why and how. What I summarized here would be taught in three different courses at my college (along with other material, of course). If you have not been introduced to all of these topics yet, this answer will contain some concepts unfamiliar to you.