Understanding of Kinetic, Potential and Change in Energy

You've asked a bunch of questions, albeit all related, so I will just try to break them down.

From what I understand, energy is defined as the ability to do work

Essentially yes. But the term capacity is generally used instead of ability.

What does this actually mean and how is it different from work itself?

Work is the transfer of energy from one thing to another. When that transfer occurs one thing loses energy and the other things gains energy. The other means of energy transfer is heat.

Also, according to my teacher, absolute energy does not exist. Why is that the case?

Generally yes, energy is a relative thing. However a zero (or near zero) energy is often assigned for reference purposes.

For example, at absolute zero temperature a thermodynamic system is generally considered to have its lowest energy as the motion of atoms and molecules of a substance comes to a stop. For an ideal gas whose internal energy is strictly kinetic, one can then assign an internal energy of zero at absolute zero temperature.

Another example is assigning a gravitational potential of zero at an infinite distance away from a gravitating body.

In steam tables for water the triple point at which solid, liquid and gas phases can all exist together is used as the zero datum for specific enthalpy, entropy, and internal energy. Then all the absolute values of these quantities given in the steam table are understood to be with respect to the triple point.

As a practical matter it is generally only the difference in energy between states that is needed to determine things like work.

If absolute energy does not exist, and if change in energy is the difference between the energy of an object at two points in time, then how can change in energy be defined, if it is in terms of absolute energies?

The change in energy can be defined by assigning a state as having zero energy, as I described above. For example, the internal energy of an ideal gas is kinetic only. By assigning a value of internal energy of zero at absolute zero temperature per the equation

$$U=nC_{v}T$$

One can determine changes in internal energy between two equilibrium states by the equation.

$$Delta U_{1-2}=nC_{V}(T_{2}-T_{1})$$

Why are terms such as "kinetic energy "and "potential energy" used sometimes instead of "change in kinetic energy" and "change in potential energy" when describing energy if absolute energy does not exist?

Because whenever the term "kinetic energy" and "potential energy" is used, it has to be used with respect to a specific frame of reference.

For example, if we say a car of mass $m$ having velocity $v$ has a kinetic energy of $frac{mv^2}{2}$ we must say what is the reference frame that $v$ is measured with respect to. Generally, that reference frame would be the road. But if two such cars were moving together side by side at the same velocity with respect to the road, the kinetic energy of each car with respect to the reference frame of the other car would be zero.

Similarly, for gravitational potential energy. If an object of mass $m$ rests on my desk a height $h$ from the floor supporting my desk, that object has a gravitational potential energy of $mgh$ with respect to the floor. But If my desk is on the second floor of my house and that floor is a height $H$ from the ground, the object's gravitational potential energy with respect to the ground is $mgH$. Once again, the energy only has meaning with respect to a particular reference frame.

Additionally, what is the specific difference between potential and kinetic energy? I understand that the former is dependent on position, while the latter is dependent on velocity.

Your understanding is correct. Potential energy is energy of position and kinetic energy is energy of motion.

However, I would like a more in depth explanation because I don't understand why the change in potential energy is equal to the change in kinetic energy of an object over two points in time.

The change in potential energy is not always equal to the change in kinetic energy between two points in time. You are probably thinking about a falling body in a gravitational field where the loss of gravitational potential energy of an object between two vertical points equals the increase in the kinetic energy of the object (neglecting air resistance).

But say I apply a net horizontal force $F$ on an object of mass $m$ initially at rest with respect to my desk over a distance $d$ the net work done is $Fd$ and the change in kinetic energy is $frac{mv^2}{2}$ where $v$ is the velocity of object after moving a distance $d$. But since the motion of the object was horizontal, there is no change in its gravitational potential energy with respect to the floor, or the ground outside.

Hope this helps.