The risk premium is the difference between the highest risk-free return available (generally that from government bonds) and the rate of return investors expect from another asset over the same period.
Investors demand high returns for higher risks so they will therefore expect to make more out of holding volatile equities than more stable bonds, for example. If government bonds are returning 3% after inflation and the real return on equities is 8%, investors in equities are earning a 5% equity risk premium.
During a bull market investors begin to feel that equities are less risky than they have been in the past. The risk premium and hence the yield that they demanded from equities shrinks as a result, and this in turn is one explanation for the rise in share prices. (In theory, if bonds return 3% and equities 8% but the risk premium falls to 3% from 5%, equity prices will rise until equities yield only 6%.)
However, the fall in risk premium eventually ends as new risk factors like deflation, corporate mismanagement, terrorism, and political instability increasingly concern investors and push prices down.