In British law, a monopoly is technically a firm with over 25% of the market. Another definition is a firm which is the market.

A true monopoly has alot of discression in the amount they are able to charge for the products , their products are effectivley exceptions to the rules of price elasticity of demand to an en extent.

Nintendo isn't doesn't have a monopoly over the gaming market , they just have alot of marketshare.

This isn't true, monopolies can only do this thing in markets which are necessities, not luxury items.

IIRC for some game theories applied to markets, market players are those with share >=25%, market leaders with >=40%, monopolists with >=71%, there was an article in an issue of IEEE's Computer gazette more than 10 years ago, I can't find it anymore, so don't ask me more, ...
There were also warnings about it: a monopoly is solid if the market has limited resources, while an expanding market and/or a bigger external new market marginalizing the older may shatter existing monopolies incorrectly taken for granted (that's why MS is so worried about linux adoption in new markets and twists and changes towards unforeseen directions to IT market previously attempted by, for example, Sun and Netscape, but failed or largely scaled down, while now Google is trying the same with much more determination, knowledge of past challengers' errors and an uncanny ability to catch MS off guard with new moves).

This is tricky. A monopoly exists when one company has sufficient share to "control" a market. Microsoft has a monopoly on business desktops in the US because the vast majority of companies pay whatever is the going rate dictated by ms... They are not in a position to bargain with ms or switch to another platform without a significant impact to their business. Ms does not have a monopoly on the home computer market because apple offers a sufficient alternative for a good percentage of users.

The same goes for servers... Viable alternatives exist, so no monopoly.

"Strictly, monopolies exist when there is a single supplier in a market. Adam Smith, in the Wealth of Nations, provided a sustained attack on monopolies but thought of them more as multi-firmed industries with statutory protection along the lines of the medieval guilds."

Hmm actualy if there is only one supplier of a good in demand then it's not really relevant if the good is a luxury or a neccasity , because the item can only be obtained from one person that individual has full discression in how much they are able to charge for it... to an extent.

Even if a item is a neccasity there's only so much that can be charged for any item  but I will agree that monoplies have more discression with neccasery items.

Monopoly = Make, Distribute & Sell Product.

^ Yes, but what you're basically saying is that luxury goods are perfectly inelastic, when they're not, they're the complete reverse.

So and if the competition just doesn't make it?

Example: Gameboy Era against Gamegear. There was market-share retained in gamegear, but it was stopped!

Sorry , I'm assuming that there is constant demand for the goods like in the games industry. I was looking at a monopoly more in relation to lucrative markets ; of course having a monopoly on a particular luxury good won't neccaserily mean that you can determine the price.