Is the Planck length the smallest length that exists in the universe or is it the smallest length that can be observed?

Short answer: nobody knows, but the Planck length is more numerology than physics at this point

Long answer: Suppose you are a theoretical physicist. Your work doesn't involve units, just math--you never use the fact that $c = 3 \times 10^8 m/s$, but you probably have $c$ pop up in a few different places. Since you never work with actual physical measurements, you decide to work in units with $c = 1$, and then you figure when you get to the end of the equations you'll multiply by/divide by $c$ until you get the right units. So you're doing relativity, you write $E = m$, and when you find that the speed of an object is .5 you realize it must be $.5 c$, etc. You realize that $c$ is in some sense a "natural scale" for lengths, times, speeds, etc. Fast forward, and you start noticing there are a few constants like this that give natural scales for the universe. For instance, $\hbar$ tends to characterize when quantum effects start mattering--often people say that the classical limit is the limit where $\hbar \to 0$, although it can be more subtle than that.

So, anyway, you start figuring out how to construct fundamental units this way. The speed of light gives a speed scale, but how can you get a length scale? Turns out you need to squash it together with a few other fundamental constants, and you get: $$ \ell_p = \sqrt{ \frac{\hbar G}{c^3}} $$ I encourage you to work it out; it has units of length. So that's cool! Maybe it means something important? It's REALLY small, after all--$\approx 10^{-35} m$. Maybe it's the smallest thing there is!

But let's calm down a second. What if I did this for mass, to find the "Planck mass"? I get: $$ m_p = \sqrt{\frac{\hbar c}{G}} \approx 21 \mu g $$

Ok, well, micrograms ain't huge, but to a particle physicist they're enormous. But this is hardly any sort of fundamental limit to anything. It isn't the world's smallest mass. Wikipedia claims that if a charged object had a mass this large, it would collapse--but charged point particles don't have even close to this mass, so that's kind of irrelevant.

It's not that these things are pointless--they do make math easier in a lot of cases, and they tell you how to work in these arbitrary theorists' units. But right now, there isn't a good reason in experiment or in most modern theory to believe that it means very much besides providing a scale.

None of the above. Though there are many speculations about the significance of the Planck length, none is proven in any currently accepted theory.

It is expected, though, that quantum gravity effects become definitely non-neglegible at the energy/distance scale set by the Planck length, so it provides a heuristic scale at which we should not expect our current theories to make accurate prediction.

There is a tiny bit more going on than the otherwise excellent answer by zeldrege suggests. Imagine that you wish to probe an unspecified object to examine its structure. If we use light to look at the structure of an object, we need to have its wavelength smaller than the size of the details we wish to look at. Probing an object that has a (linear) size equal to the Planck length, requires that the energy of the photon be greater than the mass of a black hole of that "size". So, a classical black hole would be formed by our energy probe, thus preventing us to see details inside the object we wish to investigate. We are lead to an apparent contradiction, which suggests an incompatibility between Relativity and Q.M.