In an asset sale, the buyer acquires some or all of the contents of the business such as equipment, inventory, and accounts receivable. Asset sales do not include company liabilities, and thus are typically more preferred by buyers (compared to equity/entity sales).
By contrast, in an equity sale, the buyer purchases equity in the business (or in cases of 100% of the equity, the entity itself) and thus also includes the liabilities of the company (therefore typically preferred by seller).
Also, asset sales tend to result in higher tax costs for the seller due to ordinary income treatment on some assets and potential double taxation. However, buyers prefer asset purchases for the step-up in depreciable basis.
Equity sales provide sellers with lower capital gains rates, but do not allow the buyer to increase asset basis for depreciation.