Transfer learning is a technique that is used in machine learning in general, and not just supervised machine learning. Transfer learning is a way to fine-tune some model's parameters for a specific task. Think of the model's parameters as dials, and suppose that someone already turned these dials in such a way as to optimize some criterion, such as classification error, for a specific task, such as classifying images of cats and dogs.
Now suppose that you also want to optimize the classification error and classify images of rabbits. Since someone already put the model's dials in the correct position for images of cats and dogs, you would assume that the positions of these dials would be similar for images of rabbits. Therefore, it is easier to build up on the work of someone else, rather than to start from scratch. This is the basic idea behind transfer learning.
Conversely, from Wikipedia:
Supervised [machine] learning is the machine learning task of learning a function that maps an input to an output based on example input-output pairs.
Moreover, transfer learning is a method that helps to perform supervised machine learning.