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Earthquake prediction is the art or science of determining the time, location, and magnitude of a future earthquake. In this form, earthquake prediction is protoscience at best, and is likely to be impossible. The mood was different in the past, and much research in the 1960s and 1970s went into prediction, but no prediction methods met with general success and by the 1990s research focuses had shifted. Of course, the lack of success did not deter the many visionaries who claim to have uncovered surefire means of anticipating a big one; often using techniques which were tried extensively in the 1960s and 1970s, and did not work.
In contrast, earthquake forecasting — assessing seismic hazard based on the probability of an earthquake occurring in a given region — is possible, and most attempts are reality-based. The terms mean the same thing in common usage, but where earthquake prediction attempts to warn of a particular earthquake before it begins, earthquake forecasting looks for areas where earthquakes are more or less likely.
Also distinct from earthquake prediction is earthquake early warning, detecting seismic waves after the earthquake begins, but seconds to minutes before damage occurs.
It has long been conjectured that the behavioural pattern of animals can drastically alter in anticipation of a big one. Reports of this alleged phenomenon date back to Ancient Greece. Unfortunately, any evidence of this is merely anecdotal and may simply be a result of confirmation bias. Because this idea cannot practicably be studied in a lab, it is difficult to get a scientific handle on it.
One interesting scientific analysis was an attempt by California Geology to verify a widely believed correlation between increased missing pet notices in San Francisco and earthquakes in the area. No link was found.
Although reports of seismic prescience in animals are widespread across the globe and throughout history, such reports are not easily quantifiable by science and are more than likely a crock of shit.
Animals do in fact often notice earthquakes before humans feel them, but still after the earthquake has happened. Earthquakes generate multiple types of seismic waves, the most important ones being P- and S-waves. The P-wave travels faster than the S-wave, arriving several seconds before it, but causes little to no damage and is often too small for humans to feel. Some animals can feel the P-wave, and might start to behave unusually a few seconds before the more damaging S-wave arrives and humans feel the earthquake. 
There is some evidence that stresses in rock provoke emission of radon. Geophysicists routinely measure radon concentration in well water in seismically active regions. It is likely that at least some earthquakes are preceded by changes in radon emissions, and some people have claimed successful earthquake predictions based on radon. But other factors unrelated to earthquakes can cause changes in radon emissions, so radon methods would generate too many false alarms to be useful, and it's not likely that there are radon changes associated with all large earthquakes.
The "seismic gap" hypothesis presumes that earthquakes are "characteristic" - individual earthquakes are similar, but occur at different times - and that faults are subdivided into distinct segments. An earthquake involves movement of only one or a few segments, but plate motion obviously requires movement on all fault segments. This hypothesis suggests that a fault segment which has not had a large earthquake recently, or a seismic gap, might be due for another earthquake to accommodate the plate motion. This method has been used for prediction, but in general large earthquakes do not occur preferentially in the identified seismic gaps.
Certain cloud patterns associated with impending earthquakes have been postulated as far back as 500 CE. There are currently some proponents who seem to think some mundane altocumulus cloud cover precipitates an earthquake, often many weeks in advance. It's just as meaningful to claim that a full moon is to blame as one is very likely to happen in the weeks leading up to any earthquake.
Despite countless failed prophesies, the same old escape clause gets trotted out: we prayed for mercy and the Lord saved us.
- Kagan, Yan (1997). "Are earthquakes predictable?". Geophys. J. Int. 131: 505-525. http://moho.ess.ucla.edu/~kagan/GJI_1997.pdf.
- Animals & Earthquake Prediction. U.S. Geological Survey. 2012 August 9.
- Kagan, Yan; Jackson, David (1991). "Seismic Gap Hypothesis: Ten Years After". Journal of Geophysical Research 96: 21419-21431. http://moho.ess.ucla.edu/~kagan/JGR_1991_after.pdf.
- Foster Disbelief. Wait, Oh, Nevermind, I Follow Your Logic Now….I Mean, What? Foster Disbelief. 2012 July 12.
- Mallory Simon. Global 'Boobquake' effort to disprove Iranian cleric begins This Just In. CNN. 2010 April 26.