Knots and Physics
First we need to know what a knot is. I'll refer an interested reader to the wiki page on knot theory. And I'll try to include some pictures. Knot theory is a very visual subject. But it can also be done via quantum gauge theory (Ed Witten's work for example: expectation values of a Wilson loop in 3-D Chern-Simons theory = Jones Polynomial).
We consider a circle. Then we place (embed) this circle into a three-dimensional space. Basically this means we now consider some arbitrary closed curve in this space. Knot theory concerns itself with determining whether two of these closed curves are equivalent.
This knot is not equivalent to a regular circle (unknot): there is no way of untangling it into a circle. Try it!
Knot theory in physics was first invoked by Lord Kelvin as an attempt to explain chemistry. Atoms were not known, so the reason why certain atoms bind to others and certain molecules are created was a mystery. Kelvin thought that perhaps atoms could be thought of as little knots. Molecules were then links, or simply two knots intertwined together such that they cannot be separated.
Here we have three linked unknots. Clearly Lord Kelvin was wrong about his molecules hypothesis.
But knots come back into physics in other ways. They creep into statistical mechanics, though I am not sure how yet. I will mention the topic I am more familiar with: quantum gravity.
Here is a spin network, first written down by Roger Penrose. He was hoping to create space and time from simply adding quantum angular momenta. Technically these labels on each line labels a representation of the group $SU(2)$ (for angular momentum/spin). But we don't have to think of it that way right now. We will take the dimension of the representation. If the label is $n$, the dimension is $d = 2n + 1$.
So we can "resolve" each line. Each labeled line can actually represent "d" strands going in the same direction. So if a line is labeled with a 1, we can picture 3 strands instead of the one line with a 1 next to it.
What happens at the points (vertices)? Well there are a number of different ways all of these strands can go. We simply have to stipulate that a strand cannot loop back around into the same line; it has to go to the two other lines connected to the vertex.
So we choose a way of connecting all the strands in a consistent fashion. And so we have a big mess of knotty links! Knot theory has appeared when we were simply trying to build up space using quantum mechanical angular momentum! We can use some knot theory to help put a number (amplitude) to a particular spin network. It is just a combinatorial problem that is now rendered more visual; we are manipulating strings and counting how many ways these strands can be connected.
This actually also probably comes from the Ed Witten thing, calculating expectation values of Wilson loops (this is me speculating right now). Spin networks found their way into physics through Loop Quantum Gravity. This is (quantized) general relativity where we have moved away from expressing gravity with the metric or similar objects. Gravity is now expressed by Wilson loops! Hence the name loop quantum gravity. So a spin network is just a visualization of a calculation in quantum gravity... Well, kind of. I won't get into the details (only partially because I don't know the details yet).
This whole notion of a spin network can be generalized, and it has been. Spin networks can only describe space. We need to go into another dimension if we want to describe spacetime. We will end up with things called spinfoams. Spinfoams are the basis of a large class of proposed ideas for quantum gravity (spinfoam models, group field theory, for example).
Of course, there is a big literature on knot theory and physics. And what I discussed doesn't nearly exhaust all applications. For example, people are using similar structures (actually braids, not knots & links) to try to get matter out of spin foams. And others. Where else can we get a nice interplay between knot theory and physics?