One of the three "lower" or unfortunate realms of existence within the six realms of Samsara, the animal realm is characterised by suffering, ignorance, and fear. Animals experience the same four unavoidable sufferings as humans — birth, ageing, sickness and death — yet they are plagued with ignorance and are unable to see the dukkha in this process.
They are driven by instinct and live primarily to survive while in a cycle of being preyed on and preying on other animals. Animals face intense suffering from the elements, hunger, and exploitation. The ignorance in animals also includes the inability to understand the Buddhist dharma, which makes it impossible for them to cultivate good karma most of the time, and impossible to reach nirvana.
Physically, animals exist in the same world as us but in the spiritual sense, they are a lower plane of existence and realm within Samsara. If one's karma is dominated by ignorance, greed or anger then they would reincarnate in the animal realm.
Rebirth as an animal is generally seen as a serious spiritual setback — a "bad" rebirth — where an animal is stuck in that form because it cannot create new positive karma to rise out of it. Once the negative karma that causes animal rebirth is exhausted, it is possible to be reborn as a human again, though it may take a long time.
There are exceptions, such as animals who hear the Dharma (chanting) or are well-treated by people, who can create positive imprints and thereby attain a higher rebirth later.