In simple words: AI is the broad field; ML learns from data; DL uses layered neural networks.
Start with AI
Artificial Intelligence (AI) means using computers for tasks that usually need human intelligence, such as recognising a face or answering a question.
Then ML and DL
Machine Learning (ML) is a part of AI. It finds patterns in examples instead of following only rules written by a person. Deep Learning (DL) is a type of ML that uses neural networks with several processing layers. More layers alone do not guarantee a better result.
See the flow
AI→ML→DL
Simple example
A spam filter learns from past emails labelled spam or not spam. A deep learning system might also learn the meaning of words in the email.
Remember: AI is the broad field; ML learns from data; DL uses layered neural networks.