When average cost is falling, marginal cost must be rising. At minimum average cost, marginal cost must be below average cost. In the short run, total cost and variable cost are different by the amount of the fixed cost. Average cost is the slope of the total cost curve.
Marginal cost is the slope of the total cost curve.
Marginal cost is the slope of the variable cost curve. Average variable cost and average total cost get closer together as output rises because average fixed cost gets tiny as output rises. In the long run all costs are variable. In the short run all costs are fixed. If firms minimize costs, then the marginal cost of raising output by one unit is the same whether output is increased by employing more labor or more capital.