1. What is a Lolita?
Perhaps based on the book "Lolita" by Vladimir Nabokov, a lolita could mean a youth who is not innocent. The book relates a story of a middle aged man who has a sexual relationship with a young girl (12 years old) named Lolita. To have a "Lolita-complex" means to be in love with an extremely young person. (Thanx to Juliet for the synopsis and advice.)
Although distant from the European meaning of the word, the Japanese lolita fashion is simply a style which includes gothic and/or historical dress mixed with other elements. The style is largely based on the old fashioned dolls with their frilly clothing, but is not limited to that. One of the more outstanding qualities of this Japanese fashion is that the skirt is generally shorter, from above the knee to mid-calf. Platform shoes are another distinct mark regarding this style.
There are also variations within the lolita style, a few common types are:
Gothic Lolita - a perfect mixture of gothic and lolita, it incorporates monocolors (black/white), knee-length poofy skirts, lace/ribbons and platform mary janes
Classic Lolita - a more "fluffy" style, with a variety of bright colors or even patterns and of course frills, lace and ribbon.
Elegant Gothic Lolita - perhaps a more traditional/historical style, very closely related to Gothic Lolita. I always think of a heroin in a vampire movie.
Elegant Gothic Aristocrat - a more masculine style (if that is possible) with longer skirts or wide-legged trousers, button-down dress shirts or jackets/vests.
Nothing is limited to just what I've mentioned. There are many more styles and directions one can take. Some lolitas incorporate morbid themes such as blood stains on white dresses, bandages, skulls, etc. Others choose more aristocratic accessories like a cross or a crown.
If you would like to know more about these specific GL categories, then please visit Apsara's Black Lolita site.

2. Who popularized this fashion?
The manga artist, Kusumoto Maki, was a big influence in making the Lolita style popular. In the 80s she created a manga called "KISSxxxx" which was about a rock band. One of the band members had a girlfriend who always dressed like a doll. She was considered very cute and attended all of the band's lives. Could this be why you see many of the Lolitas at lives today? (Thanx to Bronny for the info.)
Since her manga, various Visual Kei artists have used this style for their own. Mana (former Malice Mizer), in particular, has been a major influence by coining the terms "Elegant Gothic Lolita" (EGL) and "Elegant Gothic Aristocrat" (EGA) and even creating his own line of clothing, Moi-meme-Moitie. His style is inspired by old horror/vampire movies which he loves. Despite this, he did NOT create gothic lolita fashion.

3. Where/what is Harajuku?
Harajuku is a "city" in the province of Tokyo. Every Sunday, people dress in their visual best to show off their style. The Yoyogi park entrance is where people gather and parade around in either their lolita or gothic style. Some even cosplay their favorite Japanese band member.

4. Why in Harajuku? Why Sunday?
In Harajuku, there was a thing called Hoko-ten, or pedestrian paradise. To quote Shoichi Aoki in FRUiTS, "It used to be the case that cars were not allowed in the main streets of Harajuku on Sundays, and that the entire area became home to fashion designers, underground culture and creative people of every kind." Unfortunately, now this is not the case.

5. What is Cosplay?
Cosplay is the shortened term for "Costume Play" or dressing up in a costume. Many Japanese bands have elaborate clothing and make-up, making them ideal targets for cosplayers. It is truly a unique art form. The lolita style is not to be confused with cosplay, although I have heard cosplay used in reference to it.

6. What is Visual Kei?
A subgenre of Japanese Rock (J-rock), Visual Kei (kei means group) bands devote a lot of attention to their visual aspect, which for them is just as important (if not more) as their music.

7. Where can I buy?
The obvious place to buy a lolita dress would be from a store in Japan. Most of the online stores collected in my Portal section have their own offline shop. Unfortunately, I have yet to find an online Japanese store to accept overseas orders. That being the case, here are a few possible sources for finding a gothic lolita outfit. For a list of online stores, please visit my Portal section (look under the Where to Buy heading) for online shops. Please note that I have not ordered from any of these places and so I cannot guarantee their quality or service. For a list of the places I have purchased from, please check out my Shop Reviews page.
You can also take a look at:
eBay (search: "gothic") - Periodically checking eBay, you can find many different options to choose from. Remember, just because they call it Gothic Lolita, doesn't mean it is!!
Yahoo Auctions Japan (search: "gothloli" in katakana) - Yahoo Auctions Japan is better if you know some Japanese. Each item will tell you in English whether the seller will send overseas or not, but if you are serious about something you would like to purchase, e-mail the seller (in Japanese preferably) before you make a bid. To purchase from Yahoo Auctions Japan, you will need a Yahoo Auctions Japan account. (Thanx to Kakyuu for the info!)
Also offline shops such as Electrique Boutique or any other gothic shop, specialty store or costume shop. Don't be afraid to take a look in a second hand store. I even found one of my dresses at a Dillard's outlet.

8. What other things can you recommend regarding the Gothic Lolita style?
Clothing aside, there are other things that may be of interest to a Gothic Lolita. Please take a look at my Recommended Reading for information on books and magazines.
One obvious influence to the style is the story Alice in Wonderland by Lewis Carroll. His whimsical drawings may have possibly lead way to some of the more popular lolita themes, such as crowns, spades/hearts/clubs/diamonds, aprons, etc. But how much of it has really influenced the G&L style? That's still a mystery to me. Victorian Maiden's online shop includes some Alice goods. (Thanx to everyone on the Black Lolita ML for discussing it with me.)
Dolls are a lovely accessory for any lolita. Brands such as Angelic Pretty create matching dresses for person and doll. Victorian Maiden does the same for the Jenny Doll (a famous doll brand, kinda like Barbie), creating miniature brand clothing. For more variety in doll bodies, Volks is the maker of the best. Their Super Dollfies have equisite details and posable joints, completely outdoing Barbie (in my opinion) in quality and price. You can find many lovely lolita dresses on Japanese auction boards just for your doll, but beware of the prices and what comes with that pricetag. I know very little about this expensive hobby, so I recommend doing some research before committing to any particular doll. For example, some dolls need to be put together, painted, etc. Please visit the Portal page for more doll links.