Deep learning: a particular type of artificial neural network with many hidden layers (and the associated tech to make this work/trainable)

AI: The field of computer science which aims to make computers smarter. Like most fields, there is much overlap with others, for example, statistics.

Big Data: A buzzword. About the best definition I can find is anything which has the 3 V's: Volume, Velocity & Variety. In general, outside its use as a buzzword, I think big data is generally thought of as "when you need a distributed system to process your data", be it because of volume, velocity or variety.

Supervised and unsupervised learning: whether or not you require example data for training

Machine learning: some people say its the subset of AI that deals with statistical methods, other people say its just another word for AI.

Furthermore, suppose you have labels for some but not all points on your data (i.e. your model is designed to be robust in the face of things it hasn't been trained for). There are a nontrivial number of people who work on either side of the "semi-supervised" divide, e.g. clustering with examplars or pulling out the generative model for a discriminative task. Personally I like these better, as they're more akin to what people seem to actually do (encounter new things and try to make sense of them).

Anyways. If you look at the delta in performance between "old" techniques like random forests or gradient boosting vs. deep convolutional networks, it tends to be quite small until your datasets grow to very large sizes. For things like images that's not much of a problem. For things like rare diseases it's a huge problem.