A fling is when you have a short, casual romantic relationship with someone. It's usually nothing serious.
When people talk about having a fling, they're often referring to a brief and not very committed romance. For example, "I had a fling with someone over the summer" might mean that you dated casually for a few months but didn't plan on making it long-term or serious. A lot of times, people have flings when they're traveling or in new places and don't want anything too complicated.
Also, you can use "fling" to talk about throwing or tossing something with a bit of force - like, if you fling your bag onto the bed, it means you're putting it down there fairly roughly.
How common is "fling"?
Word fling is considered uncommon in modern English. It appears most frequently in fiction texts and less often in other writings.
Definitions
noun
An act of throwing, often violently.
An act of moving the limbs or body with violent movements, especially in a dance.
Example: the fling of a horse
An act or period of unrestrained indulgence.
A short romantic, oftentimes sexual, relationship.
Example: a summer fling
(figuratively) An attempt, a try.
Example: give it a fling
(obsolete) A severe or contemptuous remark; an expression of sarcastic scorn; a gibe or taunt.
(dance) A lively Scottish country dance.
Example: the Highland fling
(obsolete) A trifling matter; an object of contempt.
Example: ante 1800, old proverb
England were but a fling / Save for the crooked stick and the grey goose wing.