Machine learning and deep learning are both forms of artificial intelligence. You can also say, correctly, that deep learning is a specific kind of machine learning. Both machine learning and deep learning start with training and test data and a model and go through an optimization process to find the weights that make the model best fit the data. Both can handle numeric (regression) and non-numeric (classification) problems, although there are several application areas, such as object recognition and language translation, where deep learning models tend to produce better fits than machine learning models.
Machine learning explained
Machine learning algorithms are often divided into supervised (the training data are tagged with the answers) and unsupervised (any labels that may exist are not shown to the training algorithm). Supervised machine learning problems are further divided into classification (predicting non-numeric answers, such as the probability of a missed mortgage payment) and regression (predicting numeric answers, such as the number of widgets that will sell next month in your Manhattan store).
Unsupervised learning is further divided into clustering (finding groups of similar objects, such as running shoes, walking shoes, and dress shoes), association (finding common sequences of objects, such as coffee and cream), and dimensionality reduction (projection, feature selection, and feature extraction).