When the price is such that supply equals demand for e.g. bicycles, everybody who wants to buy a bike at that price gets one, and every bike gets sold. Everybody goes home happy and the marketplace clears. Hence the price at which the supply and demand curves cross is the Market-Clearing Price.
This isn't enough to make it an equilibrium price. "Equilibrium" means that if the price is there, it'll stay there. If it's a stable equilibrium, then even if it starts above or below that price, it'll get pushed towards it.
I feel like I understand why prices starting below the market clearing price, will get pushed up. Of course a firm will raise prices: they can charge more per unit, and sell more units (assuming a rising supply curve).
If you start above market-clearing price, there is a trade-off, because sales are demand-limited (assuming the demand curve is decreasing). Raised prices would mean more revenue per sale, but fewer sales. Lowering the price would mean more sales, but less revenue per sale. Intro textbooks (e.g. Mankiw) seem to claim that the mere fact of there being a trade-off at all, is sufficient to push prices down, but it doesn't seem like that to me. Whether lowering prices would increase profits depends on the slope of the demand curve; if it's shallow enough, then the large increase in revenue-per-sale could make up for the small decrease in number of sales.
If I'm not wrong then a firm with a monopoly on bicycles would set prices to maximize revenue, i.e. the area of a rectangle whose length is quantity and whose height is price. The corner of this rectangle would lie somewhere on the demand curve, at or above the market-clearing price. The profit-maximizing price might be one where there's excess potential supply (bikes you would have been willing to make and sell - presumably you didn't actually bother making them).
I assume that competition is what makes the difference, and somehow forces prices down to (but not below) the market-clearing price, making that the equilibrium for competitive markets. But I don't get how this works, e.g. why underselling won't stop being worth it until there's excess demand.