I assume that when you say "equilibrium" you mean the equilibrium price ...
In perfect competition, the firm and the industry have the same equilibrium price. This is because all firms in perfect competition are price takers and must, in the long run, charge the same price.
The AFC curve refers to the Average Fixed Cost of producing each quantity of goods. Fixed costs refers to costs, like rent on a building, that remain the same no matter how many units a firm produces. In that case, the AFC (TFC/quantity produced) must by definition decline as the quantity produced rises.
When we speak of market equilibrium in economics it refers to level of prices at which the quantity demanded by the customers is same as the quantity offered for for supply by the suppliers. Thus the market equilibrium has has two dimensions. (1) price, and (2) quantity sold and purchased. Please note that the we are talking about quantity actual sold and purchased. Unlike quantities demanded and quantity offered for supply, the actual quantity sold and purchased is always equal.
In a monopoly market, the entire market supply is accounted by one firm. Therefore, equilibrium point for the market and for the firm are the same. In a perfectly competitive market, individual firms have no influence on the market price as the demand curve for the firm is a horizontal line at the level of the market price. Thus same price is applicable to firm level equilibrium. However the quantity supplied by each firm at this equilibrium price depends on the cost structure of the firm. The firm can supply as much as it wishes, therefore it supplies a quantity that maximizes its profit. This occurs when the marginal cost of the firm just equals the marginal revenue. In a perfectly competitive market the marginal cost and revenue at this point are also same as the market price. Since marginal cost for every firm operating in a perfect competition is same as market price, the combined marginal cost for all the firms in a perfectly competitive market is also same as market equilibrium price.
In an oligopoly it is not possible to give a fixed formula for the equilibrium point for individual firms as it is dependent on actions of competitor firms and may change from time to time in response to changing competitive action and the competitive strategy of the firm itself.
Average Fixed Cost:
Fixed cost refers to the minimum fixed cost that a firm incurs for manufacturing irrespective of the total quantity produced. Average fixed cost is simply this fixed cost divided by total quantity produced.
Thus: Average Fixed Cost = AFC = Fixed cost/Total quantity produced.
In the above equation for AFC we see that numerator (fixed cost) is constant, while denominator (total quantity produced) is variable. Therefore, AFC reduces with increasing production quantity. As a result AFC curve, which is a graph showing AFC on y-axis and production of x-axis, is a downward sloping curve.