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, though it’s sometimes used metaphorically, as in the phrase tempers flare. And it has another sense unrelated to fire—namely, a spreading shape, as in a piece of furniture or an article of clothing.
Flair means either distinctive elegance or style or a natural talent or aptitude. It’s only a noun.
Examples
Here are a few examples of flare used well:
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-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]
A white cropped sweater bearing the midriff and high-waisted fit and flared skirt looked positively fresh. [NYT]
And these writers use flair well:
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]

