Phantaxus is right about mayfly rather than dragonfly -- and about the larval stage being much longer. Mayflies live for a year or more before they hatch into adults. Their whole lives are therefore quite long.
What about much smaller creatures? Many single-celled animals live for a very short time -- each generation may be a few days or less. Though as most them reproduce by division, perhaps in a way it's them who are immortal ( not the mythical crab...).
Many smaller invertebrates have a lifespan of a couple of weeks.
Shortest lifespan of a mammal will be some kind of shrew I think -- several generations a year. Other small mammals can live a very long time, such as some bats at up to 20 or 30 years. Likewise many small birds live only one or two years, but others perhaps a couple of decades (eg chiff-chaff).
Turtle or tortoise could be right, if only the question had been "longest"...
Some animals can dry out completely for many decades, and so "live" a long time that way -- for example, water bears and some rotifers -- I think over 100 years has been recorded. (Somehow this reminds me of Woody Allen and an excellent trad jazz soundtrack...)