How does radiocarbon dating work?
The radiocarbon method is in principle fairly simple. $^{14}\ce{C}$ is an isotope of Carbon that has a relative abundance of around $10^{-10} \%$ on earth. 

Addition by @MaxW:


  $^{14}\ce{C}$ is formed in the upper atmosphere from $^{14}\ce{N}$ so
  that the relative amount of $^{14}\ce{C}$ in the atmospheric
  $\ce{CO_2}$ is reasonably constant. Since plants get most of their
  carbon from atmospheric $\ce{CO_2}$, the relative amount of
  $^{14}\ce{C}$ to $\ce{^{12}C}$ is thus constant.


It is also not stable and decays $^{14}\ce{C -&gt;}^{14}\ce{N} + e + v$.

Now a living organism based on carbon (like us) always has a stable amount of $^{14}\ce{C}$ in his body, because he has to eat and so there is a stable balance between $^{12}\ce{C}$, $^{13}\ce{C}$ and $^{14}\ce{C}$.

Now if that organism dies, it does not eat anymore, so there is no new income of $^{14}\ce{C}$. Since that decays and the other $\ce{C}$-isotopes don't, over the years, the percentage of $^{14}\ce{C}$ is getting lower and lower at a constant rate (you can calculate that rate with the half-time of $^{14}\ce{C}$).

Now when you measure the relative amount of $^{14}\ce{C}$ in a skeleton, you know since when it hasn't eaten anymore, so you know how old it is. The method can be used around 600-50k years to the past.

There are different methods of measuring isotope amounts. You can measure how fast a radioactive decay still is and calculate from that the amount of $^{14}\ce{C}$, which is the method from Libby. 

You can also use Mass-Spectroscopy, because $^{14}\ce{C}$ is heavier than  $^{12}\ce{C}$ and $^{13}\ce{C}$ you can "easily" measure the amount of $^{14}\ce{C}$. 

There are other methods, you can look this up in Wikipedia.