Earth age uranium dating
There are many radiometric clocks and when applied to appropriate materials, the dating can be very accurate.
As one example, the first minerals to crystallize (condense) from the hot cloud of gasses that surrounded the Sun as it first became a star have been dated to 4568 plus or minus 2 million years....!! Other events on earth can be dated equally well given the right minerals.
We have also obtained a very similar age by measuring Pb isotopes in materials from earth.
I should mention that the decay constants (basically a value that indicates how fast a certain radioactive isotope will decay) for some of these isotope systems were calculated by assuming that the age of the earth is 4.56 billion years, meaning that we will also calculate an age of 4.56 billion years if we use that decay constant.
We call the original, unstable isotope (Uranium) the "parent", and the product of decay (Lead) the "daughter".
From careful physics and chemistry experiments, we know that parents turn into daughters at a very consistent, predictable rate.
Hope that helps, and please ask if you'd like more details! I think that I will start by answering the second part of your question, just because I think that will make the answer to the first question clearer.
Radiometric dating is the use of radioactive and radiogenic (those formed from the decay of radioactive parents) isotopes (isotopes are atoms of the same element that have different numbers of neutrons in their nuclei) to determine the age of something.
Radiometric dating relies on the principle of radioactive decay.
We have dated meteorites using Rb-Sr, Sm-Nd, Pb-Pb, Re-Os, and Lu-Hf isotope systems and have obtained very similar ages.
The fact that the age we calculate is reproducible for these different systems is significant.
The reason that I trust the accuracy of the age that we have determined for the earth (~4.56 billion years) is that we have been able to obtain a very similar result using many different isotopic systems.
Most estimates of the age of the earth come from dating meteorites that have fallen to Earth (because we think that they formed in our solar nebula very close to the time that the earth formed).
Because geochronologists want to measure isotopes with different masses, a mass spectrometer works really well for dating things.