Bearing in mind that no single firm within a perfectly competitive industry can influence the price of their product on the market, the only decision to be made within such a firm is the one regarding output. Competitive firms must decide what level of output they wish to produce at in order to maximize their profits.
In general, profit maximization always occurs where marginal costs are equal to marginal revenue. This is the point where the last unit produced added as much to costs as it did to revenue. Thus, if we produced just one more unit it would add MORE to costs than it would to revenue.
When an industry is relatively new, a small number of firms will supply an amount of product relatively small in comparison to the demand. Thus, the price charged by the industry as a whole will be relatively high. This will allow each firm in the industry to earn economic profits (profit maximization occurs at a point where the marginal revenue is above the average total cost for the product). The diagram above illustrates such a situation. You can see that profits are maximized at a point well above the average total cost for the product.
However, as an industry developes, more firms will be attracted to the potential profits to be earned. Thus, the supply of product will increase as new firms enter the industry. This will cause the price charged by the industry as a whole to drop. If it drops to a price where profit maximization occurs where the marginal revenue is EQUAL to the average total cost for the product, then the firm will be just "breaking even." The diagram below illustrates such a situation. You can see that profits are maximized at the point where the extra revenue earned from the last unit produced is just equal to the average total cost for producing that same unit.
As an industry continues to develop, so many firms might join the industry that they reduce the price even further. The price might in fact drop BELOW the break even point. Believe it or not, it still makes sense for firms to continue to operate at prices below the break even point, because they can at least cover their fixed costs with the money they make. It would be more expensive for such a firm to shut down, because they would still have to pay for their fixed costs, yet they would not be earning any revenue!
Thus, a firm in such an unenviable situation will continue to supply their product as long as their profits can be maximized at a point ABOVE average variable costs. In such a circumstance, the extra revenue earned from each unit is greater than the variable costs associated with making that unit.
However, if the price (and thus marginal revenue) dropped to a point BELOW average variable costs, then each extra unit produced would cost more in variable costs than it earned. Therefore, the firm would be wiser to simply stop production and just pay their fixed costs. That is why the price which intersects the point where marginal cost is equal to average variable costs is called the "shut down" point.