Equity beta = how volatile a given stock's price movements tend to be relative to the overall market's movements. Takes into account the company's capital structure (so if a company loads up on debt you will see that it tends to be more volatile than it would otherwise be as it trades in the stock market).
Asset beta = how volatile the underlying business is, irrespective of capital structure. You calculate asset beta by stripping out the capital structure impacts on the equity beta.
asset beta = equity beta / (1+(1-taxrate)*(debt/equity ratio))
An asset beta is important because you can compare companies and not have the comparison be affected by capital structure choices. What is frequently done to figure out a company's discount rate is to take the average asset beta of the company's peers and then re-adjust that asset beta into an equity beta based on what you think the right long-term capital structure is.