In a perfectly competitive market, firms are said to be price takers: since a customer can buy widgets from one producer as easily as another, any widget producer on the market faces a horizontal demand curve at the equilibrium price: if the firm tries to sell widgets above the equilibrium price, customers will simply buy their widgets elsewhere and the firm will lose all of their business. (In actual markets, of course, there is never a situation where exactly comparable goods are available just as easily from one firm as from another — the theory of perfect competition is a useful idealized model rather than a naturalistic description.)
By contrast, lack of competition in a market creates a downward sloping demand curve for a monopolist or oligopolist : although they will lose some business by raising prices, they will not lose it all, and it may be more profitable in most situations to sell at a higher price. This does not mean that monopolists are not price takers. It only says that they have the option of being either a "price taker" (at a level of output of their own choosing), or a "quantity taker" (at a price of their own choosing). They can set their own price and accept a level of ouput determined by the market, or they can set their output quantity and accept the price determined by the market. They cannot set both price and output.
A firm with monopoly power in setting prices will typically set price at the profit maximizing level. The most profitable price that they can set is where the optimum output level (where marginal cost (MC) equals marginal revenue (MR) as seen on the diagram below) meets the demand curve. Under normal market conditions for a monopolist, this price will be higher than the equilibrium price (which is the price at which marginal cost for the producer equals marginal benefit for the consumer). In the chart below the shaded area represents the profits of the monopolist. The lower half represents the normal profits that would go to a competitive firm (ignoring output losses). The upper half represent the additional economic profit going to the monopolist.
As long as the price elasticity of demand (in absolute value) for most customers is less than one, it is very advantageous to increase the price: the seller gets more money for less goods. With an increase of the price the price elasticity tends to rise, and in the optimum mentioned above it will for most customers be above one.