There are exactly two definitions of monopoly:
1. A board game that makes you hate your friends, and
2. The exclusive possession or control of the supply or trade in a commodity or service, i.e. a single seller
Single sellers (much like monopsonistic single buyers) are price makers, operating outside of the normal confines of market pressure by controlling all of the influence concerning a single good or service (all of the supply, or all of the demand). Price takers are what everyone else is - they deal with the price the market sets through supply and demand. Individual workers are price takers in that the market decides what a given wage is for a given job. If a skill set is in high demand and low supply, the wage is high because many companies are competing for that individual laborer; likewise, if a skill set is in high supply but low demand, then companies are not going to have to offer higher wage to attract those employees. Unions serve to restrict individual laborers, thereby increasing the scarcity of their product (the quantity of labor for any given skill set). This restriction is absolute when a (state protected) union is allowed to force an employer to buy labor only from them in order to return the factory to productivity, turning them into a government sponsored and protected monopoly on labor for that specific factory (or, sometimes, an entire industry in a region), protected by the use of state force.
The term "natural monopoly" deals with things that are unfeasibly large for a private solution. Mainly, this is infrastructure. Besides roads, which are *almost always free to travel, infrastructure means utilities: electric, gas, phone, cable, etc. The state has an interest in ensuring the provision of utilities (which totally conforms to libertarian ideology), but the very nature of setting up utilities in an area precludes competition from a utilitarian standpoint. The state (and its people) are NOT served by different gage wire, crisscrossing lines, different connectors and hookups, and what have you. After infrastructure is laid, and a period of (usually) government set price controls for investor payback, these natural monopoly companies are often broken up into smaller private companies by the state, after established practices and codes are made industry standard and law.
Alcoa was an "organic" monopoly in that it came about naturally, i.e. without the use of coercion or intentional tactics designed to keep competition from entering the field, but is not a "natural monopoly" by definition because aluminum production itself is not a utility. The state has a more vested interest in competitive practices that keeps the price of aluminum down, which is exactly how Alcoa got its monopoly in the first place - they increased their economy of scale in order to meet a wildly growing demand. But at any point during this period of growth, any competitor could have entered the market on a lower margin than Alcoa and provided higher priced aluminum and been able to sell it for a profit, just not as much profit. The only thing keeping Alcoa from producing more aluminum at any given time was an overlap between supply-driven price and a market that couldn't afford the increased capacity for only the discount it would have offered.