What are the 12 monkeys in the show?

As Memor-X mentions in the comments, the 12 Monkeys refers to "The Army of the 12 Monkeys" in both the film and the TV series.

However, as Q states, in the film version they are a misdirection/red herring, as the people/scientists of the future are wrong about believing they were the ones to spread the virus that begins to wipe out humanity. As it turns out, they were animal rights activists...

They decide to depart for the Florida Keys before the onset of the plague. On their way to the airport, they learn that the Army of the 12 Monkeys was not the source of the epidemic; the group's major act of protest is releasing animals from a zoo and placing Goines's father in an animal cage.

At the airport, Cole leaves a last message telling the scientists that in following the Army of the Twelve Monkeys they are on the wrong track, and that he will not return.

But in the TV version, they are not only the mysterious organization that starts the plague that wipes out humanity, but they also want to 'break down time' into a chaotic state of flux, known as The Red Forest.

The Army of the 12 Monkeys is the mysterious organization behind the plague that wipes out humanity. The organization is led by an enigmatic figure called The Witness. The ultimate goal of the 12 Monkeys is actually much larger than simply unleashing a plague and destroying the world. Their true purpose is to destroy the very fabric of time and bring about the Red Forest, using a machine called Titan.

This bigger goal, along with multiple iterations of the timeline and cycle cosmology, drive most of the plots over the course of the series.

To fully answer your question, there is no in-universe reason they are called The Army of the 12 Monkeys and most likely is homage to the film, since they kept the title too, but I would say given the nature of the TV series possibly having multiple iterations of the time line (ie: there may be a 2044 convergence in season 2, implying that there are at least "2" 2044s), that the nature of "origins" to everything could be lost, as we witness a couple of cycles breaking into new cycles (with various temporal cause and effects) as opposed to really seeing where everything may have once started...

Curious Information about the Number 12

1. The opening title sequence for the TV series shows the in-universe symbol of The Army of the 12 Monkey's with "clock hands" spiraling around the image's center. The Monkeys are arranged as "12" Clock Hours.
2. There are 12 Messengers who are created from Olivia's DNA and are used to try and "paradox" Primaries.
3. When the audience is first introduced to Titan, it resides in the year 2163. 2+1+6+3 = 12