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The basic principal of radiometric dating is that you have some radioactive element that you can measure and some event that starts a clock. Each radioactive element decays into daughter elements at a known, exponential rate, known as it's half life. The half life the is time required for half of a sample of a radioactive element to decay. The event that starts the clock must either prevent the uptake of new parent atoms, or the escape of daughter atoms.
For example, with uranium-lead dating, we look at zircon crystal. These crystals will easily accept uranium atoms in place of a zirconium atom when forming, but will strongly reject lead, so any lead in the crystal must come from the decay of a unanium atom.
With potassium-argon dating, when the mineral cools below a certain temperature, known as the closure temperature, any argon that is formed by decay is trapped. If it is heated above this temperature, argon can escape. This temperature is what starts the clock.
The other major method is carbon dating. Carbon-14 is created at a more-or-less constant rate by cosmic rays colliding with nitrogen atoms in the atmosphere. As a result, the proportion of carbon-14 to carbon-13 in the biosphere is constant. Once an animal or plant dies, it can no longer take in any new carbon-14, so this starts the clock.