Branching Brownian motion
These descriptions are usually not entirely accurate, and sometimes not at all accurate. They're just stories, not mathematical descriptions.
You've just hosted a party. You wake up the morning after and the house is a tip. You clear the kitchen table to make space for breakfast. Underneath the piles of empty bottles, pringles cans and half-eaten packets of crisps is a thick layer of grime. You put some disinfectant on a cloth and wipe the table, but you miss a patch - in fact you miss a stretch from the middle of the table to the edge. A single bacterium is sitting in the middle of the table, doing what bacteria do - moving around at random, and occasionally splitting into two bacteria, which then move around randomly and occasionally split into two.
Every day millions of your cells make copies of their DNA. Millions of cells copying millions of DNA molecules each is pretty complicated, and sometimes little mistakes happen. Usually these make no difference - that's probably one of the reasons why your DNA is such a long string of molecules, so that if there are a few mistakes it doesn't really matter. But over time these little mistakes can slowly make a big difference to your DNA.
One mathematical way of looking at these questions is to consider branching Brownian motions or branching random walks. These models involve particles (like bacteria) moving around randomly and occasionally splitting into two (or more) new particles. (This immediately looks like a pretty good model for the bacteria-on-a-table picture above; maybe it's not so clear for the DNA picture. But DNA is just a long string of molecules, and we can think of a long string of molecules as a long string of numbers, and think of that long string of numbers as a position in space, so little mistakes in the DNA mean little movements.)
Basic mathematical description
These descriptions may contain some small inaccuracies in order to keep the discussion at a basic level.
To know what a branching Brownian motion is, you need to know what a Brownian motion is. It's a function B:[0,\infty) --> R. It starts from zero, i.e. B(0)=0. It's continuous (you can draw it without taking your pen off the paper). It has independent increments - that means that if t>s, then B(t)-B(s) and B(s)-B(0) are independent, and if t>s>r then B(t)-B(s), B(s)-B(r) and B(r)-B(0) are independent, and so on. And if t>s then B(t)-B(s) is normally distributed (aka Gaussian) with mean 0 and variance t-s.
The picture above shows the beginnings of a branching Brownian motion. We start off with the blue Brownian motion; after a while we replace it with the orange and green ones; next the orange one's exponential clock rings, and we replace it with yellow and red; and so on. The picture below shows what things look like if we zoom out.
It looks like there's some kind of triangular shape appearing. One way of turning that into maths is to let M(t) be the position of the topmost particle at time t, and ask whether M(t)/t converges to a limit. It's not too hard to show that it does, and the limit is the square root of 2. Then all sorts of questions emerge. Does M(t) really look like sqrt(2)t, or is it more like sqrt(2)t + f(t) for some function f(t) that could still be quite big but which satisfies f(t)/t --> 0 as t --> infinity? Each particle is a Brownian motion, so how on earth did anyone get up near sqrt(2)t? Did they stay near the top of the triangle all the way, or did they hang around in the mass of particles in the middle and then shoot up just before time t? What do the paths of other particles look like - if I give you a function g, are there particles whose paths stay near g?
Again, these descriptions may contain inaccuracies in order to keep the discussion at a basic level. If you want a precise discussion, then you'll have to read the papers!
Let's look at the questions above.
Does M(t) really look like sqrt(2)t, or is it more like sqrt(2)t + f(t) for some function f(t) that could still be quite big but which satisfies f(t)/t --> 0 as t --> infinity?
Each particle is a Brownian motion, so how on earth did anyone get up near sqrt(2)t? Did they stay near the top of the triangle all the way, or did they hang around in the mass of particles in the middle and then shoot up just before time t?
What do the paths of other particles look like - if I give you a function g, are there particles whose paths stay near g?
We can get a bit more creative with the model. We don't have to insist that every particle has exactly two offspring: the number of offspring can be random, and might depend on the position of the particle at the fission time. Similarly the rate at which particles split might depend on their position. We can ask what the process looks like from the point of view of the topmost particle - what is the distribution of the second topmost particle for example? There's a lot of work going on in this area now.
Here's another question that I thought about for a very short while a couple of years ago. Imagine we run a BBM on a circle (ie take positions modulo 2π). Imagine time as the radius of our circle, and draw the path of each particle in black. We end up with a subset of the plane marked in black. There is a unique white component that contains (0,-a) for all small a. What is the area of this component? If we draw the path of each particle as a line (or sausage) of width delta, what is the area of the set of all white points in the plane? What if delta=delta(t) is a decreasing function of t? For example we might take delta(t)=e^(-t).
Here's a picture up to t=50, although for aesthetic purposes I drew each particle in a different colour, rather than drawing them all in black.
It may be easier to think of time as a third spatial dimension, in which case our picture looks like a cylinder.