Biological organisms (such as animals or plants) are the main examples of intelligent systems that we are aware of (excluding artificially intelligent systems, so as not to discuss whether current AI systems are really intelligent or not). Consequently, biological life is often an inspiration for AI researchers to develop AI systems.
There are numerous examples of AI systems that have been introduced (at least, partially) based on or just inspired by the biology. Here are a few examples.
Reinforcement learning is based on a similar way that animals (such as dogs or pigeons) can learn. For more details, see Sutton & Barto's book (especially chapters 14 and 15).
Artificial neural networks are very approximative models of human neural networks.
Genetic algorithms are roughly based on Darwin's theory of evolution.
Ant colony optimization algorithms (and, in general, swarm intelligence) are based on the way real ants (and, respectively, biological swarms) behave. (There is even a rap song dedicated to ants).
There are probably other examples that don't come to my mind right now. See also this question.
There are cases where AI discoveries have also helped the development of biology or related fields (such as neuroscience and psychology). For instance, Sutton & Barto (on page 4) write
Of all the forms of machine learning, reinforcement learning is the closest to the kind of learning that humans and other animals do, and many of the core algorithms of reinforcement learning were originally inspired by biological learning systems. Reinforcement learning has also given back, both through a psychological model of animal learning that better matches some of the empirical data, and through an influential model of parts of the brain's reward system.