It’s easy to confuse the homophones flair and flare, but their meanings are quite distinct.
Flare
Flare, which can be used as a noun or a verb (the verb form usually being followed by up), almost always has to do with fire—for example:
The flare will come and go in a few seconds and will get brighter than Venus in the evening sky. [Mankato Free Press]
Wildfires flare up in western valley [Times-News]
The July 4th weekend is notorious for boaters shooting off their emergency flares as a substitute for fireworks. [KHON 2]
Flare is sometimes used metaphorically, as in the common phrase, tempers flare:
Tempers flare as McHenry college board ousts chairman [Chicago Breaking News Center]
Flare-up is a common phrasal noun:
Flare-ups caused by the heat and wind on Thursday were quickly doused with the use of helicopter water buckets and water bombers, provincial officials said. [CBC News]
And there is one major non-fire-related meaning of flare, namely a spreading shape, as in a piece of furniture or an article of clothing—for example:
A white cropped sweater bearing the midriff and high-waisted fit and flared skirt looked positively fresh. [NYT]
Flair
Flair means either distinctive elegance or style or a natural talent or aptitude. It’s only used as a noun—for example:
The fair included free food, games and activities with a Western flair. [NewsOK]
Like Novotny, Streever is a scientist with a flair for anecdotes . . . [Powell’s Books]