I've worked in a number of fine art galleries over the years, and I even owned one for a while. Thus, I've seen artists approach galleries in all manners. Some send slick portfolio packages with an elegant cover letter; others email a short query with a link to their Flickr page or website (my preferred method); some call to ask if the owner might do a studio visit; and some walk in off the street and beg to make an appointment (not my preferred method.) The single biggest mistake artists make when pursuing representation isn't how they approach a gallery, but rather why they do. It's a mistake for an artist to approach a gallery for no other reason than wanting their work seen and sold. But wait, you say, isn't that the reason ALL artists approach galleries? The answer is yes, BUT there's another qualifier: the gallery must be a potential match for the artist's work. Research is key. There is truly nothing that turns a gallery owner off more than an artist who doesn't understand the vision of the gallery. Every curator chooses particular mediums or a particular subject matter, usually within a particular price range, to create their vision of the gallery. It's an artist's job to uncover that vision by thoroughly researching galleries on the front end of this process. If they don't, they run the risk of pissing off the owner or curator. Because when someone approaches a vintage illustration gallery with contemporary metal sculpture, they're wasting everyone's time. I know this advice sounds simple, but I've seen this mistake made over and over AND OVER again.
If you're an artist seeking representation the best thing to do is find a repped artist whose work is similar to yours in style, technical mastery and/or price range (though if your work costs less it could be a great advantage) then dig up their résumé or CV online. Look at the galleries they've shown in and start your research there. When you contact a gallery with your query, you've now greatly increased your odds of representation by knowing exactly why you're a good fit for their roster.
(Artwork above by Doubleparlour as shown at my gallery in 2009.)