Artificial intelligence is the field that studies how to make machines do things that — when humans do them — we call intelligent: perceive, reason, plan, decide, act.

Machine learning is the subset that learns the rules from data instead of being told them explicitly.

Deep learning is the subset of ML that uses many-layered neural networks to learn its own features.

More formally, an ML program is a function $f_\theta : X \to Y$ whose parameters $\theta$ are fit so that predictions match observed data.