The Teenaxi were characterized by their short stature, measuring about a third of the size of an average Human male. Featuring large expressive eyes and mouths, the Teenaxi had short hind legs and long arms, giving them the capability of quadrupedal locomotion.
When agitated or angered, the Teenaxi were able fighters, rolling themselves in balls and launching themselves at their target, operating in large numbers to overwhelm a much larger opponent. (Star Trek Beyond)
However, during negotiations, the suspicious leader of the Teenaxi became convinced that the Fibonans intended to use the artifact as part of a plan to "eat them" and the delegation attacked Kirk in a panic, a couple of them being beamed onto the USS Enterprise by accident during his escape. A pair of Teenaxi were later (i.e., on about stardate 2263.2) chased through a corridor aboard the ship by a couple of security guards.
Following the destruction of the Enterprise, a Teenaxi named Kevin attended Kirk's thirtieth birthday party at Starbase Yorktown. Kirk noted Kevin wore a shirt but "still no pants." (Star Trek Beyond)
Star Trek Beyond co-writer Simon Pegg compared the Teenaxi's function in the film to the Tribbles of Star Trek: The Original Series, saying, "The Teenaxi sequence with the tiny aliens at the beginning was the first thing we wrote, and it was kind of a joke. But when we came to the edit we were wondering whether it should start with the [action sequence of the] shuttle coming out of the nebula. But that didn't work. It had to start with a lightness. Our initial thinking was let's make the opening of the film a sort of lighthearted episode of the show, like 'The Trouble with Tribbles'." 
A full-scale maquette of a Teenaxi was built, and was used as reference while the movie's opening sequence was being shot. The film's visual effects artists based designs on the maquette to sculpt the fierce-looking members of the Teenaxi Delegation. (Cinefex, No. 148, pp. 75-76) "They were designed to look like they were seven feet tall. They had [...] a lot of detail in the skin and horns," commented Raymond Chen, visual effects supervisor at Double Negative, Vancouver. "We did six hero builds – the grizzly old warrior, the young buck, the muscle – each with personality." The aliens were brought to life with keyframe animation, which was based on motion studies conducted by the Double Negative animators. They used a motion capture suit to decide on how the Teenaxi should walk. Since the aliens move somewhat similar to a gorilla, the animation team used cut-down crutches to extend their arms. (Cinefex, No. 148, p. 76)
Inspiration for the essentially slapstick action where myriads of Teenaxi pour down the curved sides of their chamber towards Kirk came from pinball-style arcade games such as Pachinko. "We did rigid body dynamics simulations of Tenaxi [sic] bouncing down the walls," recalled Raymond Chen. "For the guys who were down on the ground, we relied on brute force animation, filled out with keyframe animation cycles." (Cinefex, No. 148, p. 76)