Mincing is an adjective that describes someone who is being especially dainty or refined. The voices of older women who speak as though they're young girls can be described as mincing.
Mincing is primarily used as an adjective meaning to be noticeably — and perhaps oddly or unnaturally — dainty or refined. You might remark, for example, on a lumberjack who takes mincing steps across a log, or a woman who speaks to her dog in a mincing voice. In these cases, the daintiness is unnatural — it's put on for show — or in the case of the lumberjack, so he won't fall off the log.