A small amount of neurons might already solve some problem you're having. The XOR problem can be learned by 4 neurons connected to each other.
When you want raw images or similar as an input and have it be classified into 100 classes (e.g. look up CIFAR-10 or CIFAR-100), you will need an architecture with many more neurons.
After all, ANN are simply a tool. Depending on the task, that tool might need to more elaborate. And when you have all those different possible architectures, you want a common way of naming them. Labels such as Deep Learning are simply nomenclature of talking about certain groups of artificial neural networks.
XOR (an input layer of 2 neurons, 1 hidden layer of usually 2 neurons, and an output layer of 1 neuron) is a classic toy example of an ANN.
The minimal would probably be something like a NOT gate - input layer with 1 neuron connected directly to the output layer with 1 neuron.
In practice you either use whatever size your problem needs (for small problems) or match the network size to match the RAM amount in the video cards you're using.