Role in Boss Baby
After Francis Francis was kicked out of Baby Corp and sent to live with a family, he grew up with revenge in his mind. Growing up as a vengeful businessman, he founded Puppy Co. to act as a business rival to Baby Corp. This move was successful on Francis' part, as the company became much more popular, with puppies growing more popular than babies.
It is unclear as to what the company does in general with dogs. However, adult Tim states that it was "the best pet company" in the world, which means it may have been a puppy breeding business.
By the time Boss Baby takes place, Puppy Co. is an international puppy producer (breeder) led by businessperson Francis Eugene Francis. However, the corporation is actually a front for Francis's revenge against Baby Corp.
After Francis's defeat (by ironically turning into a baby), it is unclear of what happens to the company. However, as the company's crimes were not revealed, it can be assumed that Francis’s older brother Eugene is left in control. It is unclear if Francis regained the position of CEO by the time Tim's second daughter is born (as he would be an adult by this time).