In order to understand the answer to this question fully, one must first be acquainted with the definition from the Victorian Era of the word "dandy." In this case (as you can tell by the use of the indefinite article "a") the word dandy is a noun and refers specifically to a particular male who gives too much attention to his appearance in regards to clothes, manners, and class. This word often had a positive connotation during the Victorian Era in that it was considered a good thing to have these qualities, but now the word has quite a negative connotation in that the over-exaggeration of these things is not appealing in a man. Now let's apply this definition to Dorian Gray in Wilde's The Picture of Dorian Gray.
Dorian Gray, of course, is the title character. Dorian is not concerned with anything but his own appearance (as well as the appearances of others). He is reckless, living a life of parties and opulence. He is interested in ladies such as Sybil Vane (whose name is a definite pun on her vanity) only because she is an actress and looks beautiful. Once she loses her acting talent, Dorian cares no more for her. Here is a perfect quote that shows Dorian Gray to be a definite dandy:
Those who find ugly meanings in beautiful things are corrupt without being charming. This is a fault. Those who find beautiful meanings in beautiful things are the cultivated. For these there is hope. They are the elect to whom beautiful things mean only Beauty.
The turning point comes when Dorian meets Lord Henry Wotton who speaks a lot about something called "New Hedonism" (which Dorian subscribes to) and presents a portrait of Dorian Gray in his prime. Dorian wishes for nothing more than to remain in that state of perfection found within the portrait and allow the image in the portrait to age, instead. His wish is granted and, through the years, the portrait takes on the evidence of Dorian's true, horrid personality, so much so that he kills Basil, the artist. This shows that Dorian is truly a "dandy" concerned with appearance. Even though he still appears twenty in reality, because the "appearance" of the painting changes in order to reveal truth, Dorian reacts violently.