A dialect is a variation of an official language particular to not only what geographic area you come from, but the city. For example, the United States has a "southern dialect". You can only finds the words "y'all", "ain't", and "fixin" in places like Texas and Arkanasas. Some of the popular dialects are, by country: Germany has Oberdeutsch, England has Cockney, Spain has Basque, and China has Cantonese.
Japan has about 28 dialects, most of them minor. To understand Japanese dialects, you have to understand the geography. Japan is broke up into five main islands: Hokkaido, Honshu, Shikoku, Kyushu, and the Okinawa/Rikyu Islands.
The somewhat isolation of these islands allows their residents to keep their own dialects internal without it spreading to other islands. Japanese speakers from mountainous Hokkaido of the north can differ from their southern counterparts from tropical Okinawa. (By the way, the main island Honshu is home to the most heavily populated cities in Japan, including Kyoto, Osaka, Hiroshima, Nagasaki, and Tokyo.)
Miyavi was born in Hyogo, part of the Kinki (also known as "Kansai") prefecture, on the main island Honshu, in Japan, as seen in this map, #26:
Kinki, also known as Kansai, is an amazing prefecture because it's famous for having a different dialect for nearly every city. This dialect group is called Kansai, or Kansai-ben. (-ben is the suffix meaning "dialect".) By shortening their words, contracting their verbs, and replacing some of the original Japanese vocabulary with their own, the dialect emerges with a choppy rough-sounding speech pattern. In anime and pop culture, Kansai speakers are usually associated with the tough bad-ass characters. Comedians find a real thrill in trying to mimic it. Osaka-ben is by far the most popular variation of Kansai-ben, originating in the third largest Japanese city, Osaka. Hyogo doesn't have it's own particular dialect, so they can choose to either borrow Osaka-ben or use original Kansai-ben.
In Kansai-ben, instead of saying "sugoi da ne" or something ending in "~da ne" they often will say "sugoi ya nen", and often replace "da" with "ya". "sugoi da to omou" becomes "sugoi ya to omou". "sou darou" becomes "sou yarou", and so on.
Also, the negative verb ending "-nai" becomes "-hen". For example, "wakaranai" (meaning: "I don't know") becomes "wakarahen".
Some commonly heard Kansai-ben includes:
"Okini" (meaning "thanks". In regular Japanese, this would be "arigatou".)
"maido" is another way to say "thanks".
"Akimahen" or "Akan" (meaning "no way". It's like "dame" in regular Japanese.)
"Asaz" *note: the z is not a typo. (meaning "good morning". "Asa" is the actual word for "morning", but in regular Japanese, people say "ohayou" for "good morning".)
"Meccha" (in Japanese this would be "totemo" or "chou". It's like saying "totally" or "way", as in "totally cool" or "way cool". "meccha kakkoii".)
"uchi" (this is commonly used to refer to oneself, like "watashi".)
There are many others, but these are the most common. It is necessary to point out that Miyavi himself doesn't actually speak much Kansai-ben, contrary to popular belief by overseas fans. He seems to pick and choose depending on what he's saying and the mood of the conversation. He does, however, use a lot of slang that any typical guy in Japan would commonly use. Of course, that is not normally taught in Japanese classes, but it is something one picks up from living in Japan for an extended period of time.
Final note: Although it sounds as if every person in each city speaks that dialect, that's not really true. Of course there are people in Texas in the United States who do not speak with a Texan accent. There is always a base language, like regular English or regular Japanese. There are regular Japanese speakers mingled in every city that houses a dialect, so you will find a broad array of ways of speaking there~