What is Planck's Constant and why is it significant?

Well, this is quite a difficult question. I'll try to give an answer that is not too mathematical (which I tend to do usually).

First of all (sort of historically), Planck's constant is the proportionality between light of a specific wavelength (i.e. light of a specific color) and the energy a single light particle (a photon) has. This is already quite a profound statement. Energy is usually measured in Joule, while the frequency is measured in Hertz (= 1 / seconds). That means this proportionality constant has a unit of Joule * second. This unit is what physicists call the unit of an action. For someone who does not care about the mathematics of physics, an action is quite an abstract concept. You could say it is a measure for how much dynamics a system exhibits over a time interval (precisely: It's the integral of the difference between kinetic and potential energies in a system over a time interval). An interesting fact is that your physical reality around is the one that has the minimal action that is possible.

What we can understand from that really, is that Planck's constant can be seen as being related to dynamics of a system. However, it only arises in the case of quantum mechanics. I.e. it is what separates classical physics from quantum mechanics. Planck's constant sort of restricts this action in a sense. While in classical physics the action of a system can take any value whatsoever, in quantum mechanics you are always restricted to multiples of Planck's constant. In this way physicists say that classical physics can sometimes be recovered from quantum mechanics, if we assume Planck's constant to be zero (this is really only a thought experiment, we cannot change Planck's constant of course).

Planck's constant being related to dynamics of a system, it has a say in what kind of positions and momenta (that is velocities) particles in quantum mechanics can be.

Planck theorized that there is a minimum "resolution" to frequencies and energy. Through both experimentation and theory, he realized that all the frequencies and energies radiated were multiples of a single number, which came to be called Planck's constant. To simplify, you could emit at say, 10000 Planck's constants, and at 10001, but not at 10000.5.

In fact, Heisenberg's uncertainty principle says that position and momentum of a particle are related such that one cannot measure both at the same time better than Planck's constant, i.e. the product of the momentum uncertainty and position uncertainty needs to be larger than Planck's constant. This in effect means that if you measure one of the two very well, the other needs becomes more uncertain (as in actually will take values of a larger range). It kind of means if you try to trap a particle in a very small volume, it's uncertainty in velocity and direction will become huge and vice versa, because the product of the two needs to be larger than Planck's constant.

So, in a way one can argue that Planck's constant really is a fundamental unit of our Universe; our Universe is not continuous, but rather grid like on extremely small scales (heck, Planck's constant has a value of 6.63 * 10-34 Js, which is so ridiculously small I don't even know how to give a proper example). And the size of these blocks is directly proportional to Planck's constant.

Well, I hope this was somehow understandable or even answers what you want to know. This really is at the core of most of physics, so a proper explanation is always going to be lacking in some respects. Here's a video by PBS Space Time. If you have more specific questions, just ask. :)