Burn-in is intended to give the Markov Chain time to reach its equilibrium distribution, particularly if it has started from a lousy starting point. To "burn in" a chain, you just discard the first $n$ samples before you start collecting points.
The idea is that a "bad" starting point may over-sample regions that are actually very low probability under the equilibrium distribution before it settles into the equilibrium distribution. If you throw those points away, then the points which should be unlikely will be suitably rare.
This page gives a nice example, but it also points out that burn-in is more of a hack/artform than a principled technique. In theory, you could just sample for a really long time or find some way to choose a decent starting point instead.
Edit: Mixing time refers to how long it takes the chain to approach its steady-state, but it's often difficult to calculate directly. If you knew the mixing time, you'd just discard that many samples, but in many cases, you don't. Thus, you choose a burn-in time that is hopefully large enough instead.
As far as stability--it depends. If your chain has converged, then...it's converged. However, there are also situations where the chain appears to have converged but actually is just "hanging out" in one part of the state space. For example, imagine that there are several modes, but each mode is poorly connected to the others. It might take a very long time for the sampler to make it across that gap and it will look like the chain converged right until it makes that jump.
There are diagnostics for convergence, but many of them have a hard time telling true convergence and pseudo-convergence apart. Charles Geyer's chapter (#1) in the Handbook of Markov Chain Monte Carlo is pretty pessimistic about everything but running the chain for as long as you can.