Carbon dating
Life on Earth is based on carbon - the main constituent of all animals and vegetation. The "normal" type of carbon (the common isotope) is carbon-12. In addition, there is a radioactive isotope that exists in small concentrations called carbon-14. As with all radioactive isotopes, carbon-14 is unstable and breaks down, such that half of the carbon-14 decays away every 5,730 years. So, if you started with 1g, after 5,730 years you would have 0.5g, after 11,460 years 0.25g, after 17,190 years 0.125g, etc.
As animals and vegetation respire, they absorb some carbon-14 from the atmosphere in such a way that there is a "carbon equilibrium" established with the air. This means that the ratio of carbon-14 to carbon-12 is the same in the organism as it is in the air.
The basis of carbon dating goes like this,
  • When living things die, they stop respiring (nothing contentious with this!)
  • The radioactive carbon-14 in the dead thing starts to decay
  • If the carbon ratio in the remains of the dead thing is measured, then the date it died can be estimated

After about 50,000 years, there is such a small amount of carbon-14 left in the remains that it is too low to detect - hence, carbon dating can only be used for dates up to 50,000 years or so.
This sounds fine, and may work reasonably reliably for short time frames (hundreds of years), but there are two main assumptions that cannot be scientifically verified over thousands of years,
  • That the ratio of carbon-12 : carbon-14 in the air has remained constant
  • That the decay rate has remained constant

Yet we know that the ratio can change significantly (e.g. by volcanic activity, solar radiation, and changes in the Earth's magnetic field) so it is highly unlikely that recent conditions are representative or can be valid for all time. Also, if there was a global flood in Noah's time then this catastrophic event would have affected the geology of the Earth significantly.
Due to the common presuppositions within the scientific community, analysis of carbon-14 is not normally performed on remains that are believed to be millions of years old as it is generally assumed that none will be present. However, if the test is carried out, carbon-14 is often found where it shouldn't be (see further reading below).

Further reading: