How radiometric dating works
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 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.
A mass spectrometer is an instrument that separates atoms based on their mass.
For an example of how geologists use radiometric dating, read on: A geologist can pick up a rock from a mountainside somewhere, and bring it back to the lab, and separate out the individual minerals that compose the rock.
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.
Answer 2: Yes, radiometric dating is a very accurate way to date the Earth.
We know it is accurate because radiometric dating is based on the radioactive decay of unstable isotopes.