Why can't we find the exact position and velocity of a particle? And why is it that if the uncertainty in position is very, very large then the velocity can be determined, and vice versa? Please explain to me.
If you imagine particles to be like billiard balls (the way you're taught in school) then this of course makes no sense. The only way you can understand it is to accept the quantum mechanical description of particles: that they are probability distributions that exhibit both wave-like behaviour and particle-like behaviour depending on how you measure them.
Once you accept that a particle is best imagined as a probability wave smeared out through space then the Heisenberg uncertainty principle is a rather simple mathematical result relating to Fourier transforms. The more concentrated some function $f(x)$ is, the more spread out its Fourier transform $f(k)$ is. And since position and momentum space are Fourier transforms of each other (see here) then this gives rise to the uncertainty principle.
In other words, if you measure the position very precisely (highly peaked $\psi(x)$) then the momentum representation $\psi(k)$ will become very broad (imprecise), and vice versa.
It's not that you cant measure it -- it's that particles just don't have well defined positions and velocities at the same time.
Actually, I believe the uncertainty principle actually refers to the position and momentum of a wave. But I don't want to split hairs.
Now, particles, until they interact with something, appear to exist as probabilities, (or I prefer to think them as "possibilities", it's more "magical" to me, but anyway...), and their probability functions, the areas where the particles might pop up.. for some reason or another they act like waves.
The think is, with a wave, how do you get the exact position? A wave can take on many forms. It can have just one peak, or it can have multiple peaks and troughs. Now, it it has just one peak, it's position is pretty well defined, at least, if you ask me. But if it has a bunch of peaks, it's position is more "spread out".
Now, if all you have is a glimpse in one instant, the more "spread out" the wave is, the easier to be to get an idea of where it's going. If the wave is all clumped up into one wavelet, it will be really hard to tell where it wants to go. But if it's spread out, like ripples in a pond, and you happened to get a glimpse, you can kind of tell where the wave is going, you will be able to tell that it's spreading in a certain direction, from the studying the ripples you will be able to guess it's speed, etc. But what is the exact position of the ripples? How do you define the position of .. "ripples"? Yeah, you can make up a suitable definition for the macroscopic world, but in the quantum case, the waves aren't actual, physical things. They are waves of "probability", and the actual particle, the actual physical thing we are interested in, well it's busy chilling out somewhere in these probability waves, at least until it decides to manifest itself.
So basically, the way it works out, the more spread out these probability waves are, the easier it is to guess where the particle wants to go. The more you know about where the particle wants to go, then the better you can define it's velocity.
However, if the probability wave is bottled up into a small spot, you have a good idea of where the particle is likely to be in that instant, but you will have no idea of where it wants to go. To know where it wants to go, the probability wave has to be smeared out. But then you lose the information of where it is, presently.
And that's just the way it is.
You ask "Why can't we find the exact position and velocity of a particle" - this can not be directly answered because there is no particle in this sense you use it.
The thing does not have an exact position and velocity - it just is not a particle in the "normal" sense.
The fact that things on this very small scale do not have a well defined location and velocity at the same time is just what we found out how the universe is like.
With this, the question becomes similar to "Why is there gravity?".
That's a very valid question too - but a different kind of question.