Formal v Informal
A formal garden is based on geometric shapes, whether that be circles, rectangles or squares etc. They may contain elements such as stone or topiary spheres or cubes as 3-dimensional interest and there will often be a high degree of symmetry, although asymmetry can be built in through creating different axes of the garden.
An informal garden is looser and generally uses only natural materials. The lines are softer and more fluid and the planting is less rigid.
Most gardens, particularly larger ones, feature elements of both the formal and informal styles. As the house tends to be geometric in nature, the area closest to it lends itself to a more geometric approach. As you progress further out from the house the garden can become more informal.