Kuaui is officially spelled "Kaua`i" with the backward apostrophe indicating a glottal stop (the way air stops in the expression "oh-oh!"). A definitive pronunciation for Kaua`i is harder to find but most sources say, "ka-wa'-ee" with the glottal stop between the wa and the ee. You'll often hear it pronounced "ka-wye'" (with the accent on the second syllable) although this would not match its official spelling.
It is spelled "Kuaui" on this page only as a convenience to those searching with this common miss-spelling. Elsewhere on this site it will be spelled Kauai.