Artificial neural networks (ANNs) are mathematical constructs, originally designed to approximate biological neurons. Each "neuron" is a relatively simple element --- for example, summing its inputs and applying a threshold to the result, to determine the output of that "neuron".
Several decades of research went into discovering how to build network architectures using these mathematical constructs, and how to automatically set the weighting on each of the connections between the neurons to perform a wide range of tasks. For example, ANNs can do things like recognition of hand-written digits.
A "biological neural network" would refer to any group of connected biological nerve cells. Your brain is a biological neural network, so is a number of neurons grown together in a dish so that they form synaptic connections. The term "biological neural network" is not very precise; it doesn't define a particular biological structure.
In the same way, an ANN can mean any of a large number of mathematical "neuron"-like constructs, connected in any of a large number of ways, to perform any of a large number of tasks.