You can call it an artificial neural network (ANN) as soon as you have two or more artifical neurons connected to each other. As simple as that.

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.