The Association of Zoos and Aquariums defines an ambassador animal as "an animal whose role includes handling and/or training by staff or volunteers for interaction with the public and in support of institutional education and conservation goals." Put more simply, these are the animals which interact with visitors.
Sometimes they interact with visitors from within their enclosure - a lorikeet feeding aviary, a stingray touch tank, a giraffe-feeding station. Sometimes they interact with visitors on zoo grounds - you might see a docent walking around with a hawk on a glove, or a snake coiled around their hand. Sometimes, they interact off-grounds, visiting schools, community centers, and - every spring - the halls of power itself, mingling with members of Congress during the AZA's Zoo and Aquarium Day on Capitol Hill.
Animal ambassadors are trained (or, in the case of some species, such as insects, merely habituated) members of the collection. It is their role to be used to help educate visitors about the zoo's residents and its mission. Perhaps they will demonstrate unique adaptations (a kinkajou's prehensile tail, a porcupine's quills) or behaviors (birds in a free-flight demonstration). In many cases (except for animals that interact with visitors in their enclosure), these are a special class of zoo animals which live separate from the rest of the collection, usually in off-exhibit quarters. They must be prepared to deal with a variety of situations and environments that other collection animals would not be exposed to, from elementary school auditoriums to county fairs to television studios.
Not all animals - either as species or as individuals - are well suited to this lifestyle. Over the years, I've trained several animals for use in programs such as these. Some have been highly successful and have gone on to develop local rock-star status as animal stars. Others, I've had to throw in the towel on and admit that it just wasn't working out - maybe it would have with a different trainer, or a different individual of that same species, but that particularly combination wasn't going to work, and I wasn't going to unnecessarily stress-out an animal that wasn't ready to thrive in those settings. Some animals will work very well for some trainers or handlers, not for others.
The dark-twin of ambassador animals - which I reject entirely - are petting schemes. Which is not to say that all contact is bad - several animal ambassadors can be touched by members of the public in a controlled, supervised manner that makes sure that the animal isn't unduly stressed. What I refer to is plopping a baby big cat or monkey in a succession of laps for paid photo-ops. Again referring to AZA's policy statement, the goal of ambassador animals is to further the message of the zoo or aquarium - not just to create an epic selfie. Every zoo and aquarium has it's own policies of what contact is allowed and under what circumstances.
An ambassador animal can be anything from a Madagascar hissing cockroach, sitting idly in a presenter's hand, to a cheetah, let off its leash to sprint for a demonstration. Used properly, they are an extraordinary asset to a zoo's educational programming, and can turn an informal chat about animals into a lesson that visitors will never forget.