All radioactive isotopes have a characteristic half-life (the amount of time that it takes for one half of the original number of atoms of that isotope to decay).
By measuring the parent isotope (radioactive) and the daughter isotope (radiogenic) in a system (for example, a rock), we can tell how long the system has been closed (in our example, when the rock formed).
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.
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.