The marginal cost of production is the increase in total cost as a result of producing one extra unit. The concept of marginal cost in economics is similar to the accounting concept of variable cost. It is the variable costs associated with the production of one more unit.
Marginal costs are not constant. For example a factory may be operating at the highest capacity it can with all workers working normal full time hours, so increasing production by one more unit would mean paying overtime, so the marginal cost would be higher than the current variable cost per unit.
Conversely, an input may become cheaper as the quantities purchased rise (e.g. quantity discounts), so marginal costs may fall as production increases.
Long run marginal cost is the marginal cost of increasing capacity over a period long enough that there are no fix restrictions on output (e.g. extra workers can be employed so overtime rates are not paid, machinery can be bought, etc.).
The importance of marginal costs vary greatly from industry to industry, and from product to product. The marginal cost of manufacturing jewellery is likely to be high: the materials and skilled labour needed are both expensive. On the other hand the marginal cost of producing software or recorded music is negligible.
The concept of marginal cost is very important in areas of economics such as analysing optimum levels of production for a firm. Profit maximising output is achieved when marginal cost equals marginal revenue. Selling prices are either constant (given perfect competition), or fall (the usual situation where the firm has some level of market influence). Long run marginal cost usually initially falls as a result of economies of scale, but eventually rises as a result of diseconomies of scale, and increasing demand pushing up prices of inputs.
Short run marginal cost depends on particular circumstances. For example a factory that needs to pay overtime and buy supplies that are more expensive when ordered at short notice will have high short run margin costs, on the other hand a business with high fixed costs and extra capacity (e.g. a hotel that is not full) will have very low short run marginal costs up to the point where it operates at capacity.