Complementary therapy does not have a precise definition, and much depends on the beliefs of the person using the term. It is often referred to as "Complementary and Alternative medicine" (CAM). One definition would be any kind of approach that is not a recognised treatment within the field of Western scientific medicine - conventional mainstream medicine and health in Western societies.
One might also subdivide it into "Complementary health" - treatments that Western medicine tolerate or accept as possibly beneficial (eg. massage), and "Alternative medicine" - treatments that Western medicine consider to be bogus (eg. homeopathy).
The context of this is the Western medicine predominantly (some would say 'exclusively') focuses on illness (pathology), and does not address well-being in a holistic way. So Western medicine is happy to give you a tablet when you have a recognised condition, but doesn't know what to do when you are feel unwell for reasons that don't fall into a neat scientific category and doesn't care about your overall level of happiness and sense of well-being.
You might then also define convention medicine as being concerned with illness, and CAM as being more focussed on holistic well-being.