An oligopoly is a situation in which only a few companies have a control over a part of the market. There is some competition, yes, but it is dominated by a relatively small number of businesses who may even then decide to come together in a merger or a trust. This is often bad for the consumer, because this means that there's almost no one a consumer can turn to for a better deal, and if the companies decide to work together in some manner, then there's nothing the consumer can do to affect the invisible hand.
A monopoly is when a single company or corporate entity has control of the entire market for one reason or another...
A natural monopoly comes about when it would be counter-intuitive and inefficient for there to be multiple businesses working in the same industry. Such as with power or water utilities, it wouldn't be in the best interests of everyone to have multiple companies working to create multiple overlapping infrastructure grids. This often has an advantage to the consumer in that the production costs are fixed, so the more consumers there are, the lower the price for everyone because that production cost is being subdivided among a greater and greater market.
A government monopoly is one where the government owns the only real competition in the market because no other private entity would want to step in due to the lack of profitability, or because the service could not be provided due to market failures. One such government monopoly is the United States Postal Service, though modern courier services and online services are starting to chip away at this monopoly. For consumers, these monopolies help to fill a niche in the market which would otherwise go unfulfilled, though it does mean they have little choice in the matter to affect the invisible hand.
Geographic monopolies are those created by the land regions and placement of the business- there's no one close enough to compete, or it would not make sense for there to be competition in the region. If you're the only gas station for miles, you're going to naturally be in control of all the business in that area. For consumers, this is less than ideal because you're basically being dictated prices without anyone else to turn to. For producers this is perfect, because it means you can set any price within reason and know people will basically have to pay it.
Technological monopolies come about when a single company has control of the market due to its production techniques which allow it to create products either vastly cheaper than the competition, or which unique utility and incentives. Intel has a pretty decent technological monopoly at the moment with its 14 nanometre chipset production; Skylake.
A less than "true" monopoly; monopolistic competition occurs when there is not a standardized product. If a single company has a slightly different product which is more attractive than another, then consumers will likely go to them instead of another producer. This is good, because it means that businesses compete to innovate and provide features for consumers that would otherwise not exist.