Synastry is actually the term used to refer to chart comparison of any kind, but it has come to be known as the specific technique of overlying one chart's planets onto another chart, and seeing where those planets fall into the other chart's houses. It also includes comparing how each chart's planets interact, where they touch and spark each other through aspects. This technique is invaluable for understanding how each person triggers the other, what each of you can learn from the other and teach the other, and what areas of life will be lit up by each of you, for better or worse.
The Composite method is done by taking two charts and making one new chart from the joining of the two, to represent the relationship itself. This is done by midpoints. A midpoint is just what it sounds like - the middle point between two planets. This can be done in an individual's chart, but it's especially useful between the same planet in two or more people's charts. For example, if person 1 has Mars in Libra and person 2 has Mars in Pisces, then the composite Mars would probably be somewhere in Sagittarius. The Composite chart represents the relationship as an entity itself and the chart can tell us about the life of that relationship.