Most accurate radiometric dating method
The ratio of the parent to daughter then can be used to back-calculate the age of that rock. The reason we know that radiometric dating works so well is because we can use several different isotope systems (for example, Uranium-Lead, Lutetium-Halfnium, Potassium-Argon) on the same rock, and they all come up with the same age.
This gives geologists great confidence that the method correctly determines when that rock formed.
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.
They can then look at a single mineral, and using an instrument called a mass spectrometer, they can measure the amount of parent and the amount of daughter in that mineral.
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.
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).
Obviously, if the substance you are measuring is contaminated, then all you know is the age since contamination, or worse, you don't know anything, because the contamination might be in the opposite direction - suppose, for example, you're looking at radio carbon (carbon 14, which is produced in the atmosphere by cosmic rays, and which decays into nitrogen).