To begin with, the spelling is "martini." It ends in an "i." To a English native speaker, it is obviously an Italian surname. It must have been named after somebody named Martini, but nobody knows who it was or how it was invented.

A martini is a mixture of gin and vermouth. For some reason, drinking straight gin was or is considered somewhat low-class, the kind of drink you order when you just want to get drunk. As a "mixed drink," a martini is considered more sophisticated. 

Vermouth is usually a sweet aperitif. 

When speaking of alcoholic drinks, "dry" is the opposite of "sweet."

"Stiff" is an direct request for a strong drink, one with a lot of alcohol. If someone says "I want a stiff bourbon" it means there's no pretense, he wants a strong drink.

Martinis can be mixed with various proportions of gin and vermouth. Six parts gin to one part vermouth is one standard. However, proportions vary. 

There are various jokes revolving around the idea of someone who really just wants to drink straight gin but doesn't want to admit it, and therefore orders a martini but directs the bartender to use very little vermouth. "Make me a martini, but just wave the bottle of vermouth over the shaker."

So, finally, a "dry martini" is a way of saying you want almost all gin, very little vermouth without emphasizing the strength of the drink. A "stiff martini" is a way of saying the same thing while admitting that you want the drink to be as strong as possible.

I think I have to mention Robert Benchley's old joke: "Why don't you get out of those wet clothes and into a dry martini?"

Well, Dan!

I just can't express how gratefull I am to you for the explanation! You've been very helpful, thanks thousands of times!