Now I can only give a guy's perspective on dancing; however, I will be be able to give an observation of the female perspective. A lot of guys do not particularly like dancing. The reason many of them do dance is to be close to a female. For men this is the beginning of a more intimate journey. One that could go in many different directions. Granted that these guys want to begin a journey (the destination is different for everyone), but getting started on the right path is very difficult.
Many guys feel like "creepers" when they try and dance with a girl. It is a very bad idea to just walk up behind a girl and begin dancing with/on her. Most girls tend to stay with a group of friends, and once they feel like they are being creeped on, a secret code of eye movements and head nods is used causing her friends to take her away saving her from what she feels is an uncomfortable situation. It takes some ground work for a girl to feel comfortable straying from their group or at least adding another person into it.
I have an Australian friend here at Purdue who was entirely perplexed by the grinding situation at an American party. he describes grinding in Australia as a prelude to a "good night". While here in America dancing can literally mean nothing and just be a fun way to hang out. He said he felt like these girls would slap him in the face if he danced with them like that. I explained to him how he should go about it:
I said there are a few key steps that you should practice and utilize. When you're walking through a room make eye contact with some girls that you are interested in. If they return the eye contact that is step one. Next wander over to them maybe strike up a small talk conversation, otherwise make eye contact again and begin to dance around/as a part of her group. Laughing, smiling, and more eye contact means that you're doing well.
After a bit of dancing from a distance make your way closer. Don't just get up behind her; she makes the final decision. Once you get close enough to look as though the two of you are together, but aren't touching she will either go the rest of the way or back away. If she goes the rest of the way you're good to go. Have fun! If not, it's no big deal. Don't push the issue just move to another room and try again.
Now he did put this advice to use one night and came back with a wonderfully funny success story, eventhough he did say that he still felt awkward and wanted to go back to his personal space. Most girls love to dance, most guys love when they do. However, there is a process in order for the journey to be successful and for the awkward creeper feelings to be at a minimum. Have fun dancing!