See Counterexamples to an Old Earth
Old Earth theories propose that the Earth has existed for billions of years. These theories were largely developed by British and European geologists in the 18th and 19th centuries and led to the development of the Theory of Evolution, whose postulated changes could not have possibly occurred in the thousands of years that people had previously estimated the age of the Earth to be.
A principle basis for Old Earth theory is radiometric dating. This rests on the assumption that radioactive decay rates have always been constant, which is disputed by Creationists. Notice the circular reasoning in basing Old Earth assertions on the assumption of constant radioactive decay: that assumes what the proponent of the theory attempts to prove.
The discipline of uniformitarianism is also a basis for assuming an old Earth. Extrapolating the rate of geologic processes backwards in time yields an Earth substantially older than the Biblical 6,000 years. Uniformitarianism is also based on assumptions, however; namely, that many geologic processes were acting at the same or similar rates for the whole history of the Earth. Other theories held by Old Earth proponents include punctuated equilibrium, which suggests gradual change over time with periods of rapid change.
Objections to Old Earth Theory
Theories of an Old Earth frequently rely on the assumption that physical laws, such as rates of decay, have forever been constant. This assumption is circular and considered false by Young Earth Creationists.
Physical origin of the Earth
Those who accept an old Earth generally believe that natural processes formed the Earth and solar system over a long period of time. This is as opposed to the young-Earth creationist belief in an essentially instantaneous creation.