The acid scene in "Suicide Squad"

It was acid. Just not a melt your face off type acid. It bleached their skin and discolored their hair and clothing. It likely also affected them mentally, but it's debatable if it made them crazy or just proved that they were. It is implied to be the same Ace chemical factory that the Joker is said to have fall in to make him as pale green as we know him.

The significance of the scene is to establish that Joker maybe somehow also actually cared about Harley instead of just toying with her. His face before jumping down seems to show some internal conflict, as of he himself disliked that he felt something for her, which pushed him to save her against his normal desire to kill people.

It's also to establish how Harley became like she did, her look. That she voluntarily became like that, out of a type of mad love for Mistah J. Unlike her recent comic reboot where he threw her into the acid against her will. The comics make him rapier than the movie.

Then again, as this is a flashback of an "unreliable narrator" like batshit crazy Harley, we don't know if that's an accurate flashback. For one, she wouldn't have been able to see what Joker does after she fell. Line of Sight issues.

That scene is homage to her counterpart from The New 52 reboot of DC comics:

Following DC's 2011 relaunch of its titles, Harley Quinn's costume and appearance is fully revamped. Harley Quinn has a more revealing costume, altered hair color, and bleached skin. Her hair is half-red and half-black, like the jester cap of her previous incarnation, rather than fully blonde. Consistent with her new origin, her bleached skin is the result of being kicked into a vat of acid by the Joker. - Wikipedia

Joker also got his infamous look by similar incident/accident :

Ace Chemicals is a fictional business organization in the DC Comics universe. It was at Ace Chemicals plant where the Joker (when he was Red Hood) fell into a vat of chemicals, creating his trademark look of green hair, chalk white skin, and red lips. - Wikipedia

That Joker origin story is from Batman: The Killing Joke one-shot graphic novel. But Joker serves as an unreliable narrator in it.

He admits to his own uncertainty, as he has disparate memories of the single event ("Sometimes I remember it one way, sometimes another ... If I'm going to have a past, I prefer it to be multiple choice!") *

So that scene was inspired by comics and do seem non explanatory in the film. As per rumors, that scene was supposed to be a bit longer.

Refer this link for whole list of rumored cut scenes.