Volume is the total amount of shares traded over a specific time period.
Forget the geometry definitions of volume (is it length x height x width or is it width x height x circumference?). In stock investing meaning, volume is a measure of the liquidity of the company's shares.
There are two types when we talk about volume, here: share and dollar. Depending on the the interest in the company, the volume can vary from zero to many millions of shares trading on average per day. One common time frame is the average daily volume over the past three months. This is often displayed on websites such as Google or Yahoo! Finance.
Volume can also be measured on a dollar level. In this case, it is the average number of shares traded times the average price.
The more shares which trade hands, the more liquid a company's stock is said to be. Some companies have their entire float traded within a few days and have the liquidity of water. (This doesn't mean that all the shares in the float have traded, just that the number of shares traded equals the float. This implies that some specific shares are traded often, while others aren't.) While others might see several years go by before the number of shares traded equals the float. You could say that those companies have the liquidity of a brick.
If you think that you might need to get out of a position quickly, then you should look for a high volume, liquid stock.