Homeopathy is a system of medicine that based on the Law of Similiars, which works on the principle that like cures like.

Let's look at an example.  Cutting an onion commonly causes runny eyes and nose.  The remedy allium cepa (made from red onion) is used homeopathically to treat colds and hayfever if the symptoms include streaming eyes and nose. So you can see that a substance which causes symptoms in a healthy person can remove those symptoms in someone who is sick.

Modern homeopathy was established over 200 years ago by Samuel Hahnemann, a physician living and working in Germany. Becoming disillusioned by the medical practises of the day (blood-letting and leeching), he earned his living as a translator.

It was whilst translating the works of a Scottish physician, William Cullen, that Hahnemann read about the drug Cinchona (quinine) which was being used to treat malaria. Hahnemann experimented by taking some of the drug himself and within a few days he developed the symptoms of malaria.  When he stopped taking quinine, the symptoms subsided.  

Hahnemann experimented with a range of substances which were commonly used as medicines in the 18th Century.  He and his followers found that they caused the symptoms in healthy people that they were used to cure in the sick.

Many of the substances used in homeopathy are poisonous in their original form, and over the subsequent years Hahnemann developed a method of diluting substances whilst shaking and banging them (known as succussion) in order to make the remedies safe to use.  Remedies are still made in the same way today.