No backstory exists for Clover, all that is known is that it was a 350 foot monster that lived in the ocean and was awakened by a falling satellite dish. The protagonists of the film were unfortunate to have encountered the monster's attacks four times.
Clover rampaged through New York and killed any human it came across, worse a number of giant parasites were released from its body and attacked the humans as well. It even went as far as to rip the head off the Statue of Liberty and toss it into New York.
At the end of the movie, the U.S. Air Force dropped a bomb that killed all the parasites, though it is suggested that Clover survived. It couldn't have been a bomb that completely obliterated the entire area Clover was in, since the video camera which had footage of what the human protagonists experienced was not destroyed. This could most likely explain why audiences believe that it is more likely the monster survived than had been killed by the bomb droppings, although J.J. Abrams stated that the monster was killed.
The screen time of Clover was kept to a minimum, with the humans and giant parasites taking up most of the screen time. Much of Clover's rampage happened off screen with the audience witnessing the shocking aftermath.