The book value of an asset is the accounting value of the asset - the asset's cost minus its accumulated depreciation. The book value of a firm, on the other hand, is equal to the dollar difference between the firm's total assets and its liabilities and preferred stock as listed on its balance sheet. Because book value is based on historical values, it may bear little relationship to an asset's or firm's market value.
In general, the market value of an asset is simply the market price at which the asset (or a similar asset) trades in an open marketplace. For a firm, market value is often viewed as being the higher of the firm's liquidation or going-concern value.