Khaleegy means "Gulf" in Arabic, and in belly dance, it's used to describe music from the Persian Gulf region. Like any Arabic word, you'll see it converted into English in a number of different spellings - Khaleegi, Khaleeji, Khaleejy, Khaliji and so on.
There are many different styles of dance within the Khaleej region, but belly dancers use the name "Khaleegi" to describe a folkloric dance by a group of women, traditionally done at weddings. The women would throw a "Thobe Al Nashal" (what we call a Khaleeji dress) over their party clothes for the dancing.
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Featured image of Chani and Anara by Nico Nelson