Homeopathy is controversial because of its use of highly dilute medicines. For the homeopathic practitioner the main problem is to find the medicine that matches the disease and the type of patient. But from the scientific viewpoint the most challenging aspect of homeopathy is the use of so-called 'ultramolecular' dilutions.
Homeopathic medicines are made by a process of serial dilution and succussion, sometimes known as potentisation, (succussion means vigorous shaking), in steps of 1:10 (denoted x) or 1:100 (c or cH). Matter is composed of particles (atoms and molecules), so that if you dilute a substance enough, you will eventually dilute it out altogether .
The homeopathic medicines sold in most pharmacies in the UK are in 6c and 30c dilutions. While the 6c dilution is likely to contain molecules of the starting substance, it is extremely unlikely that a 30c dilution does. Such dilutions are called 'ultramolecular'.
This is a problem! But recent scientific work suggests that it may not be an insuperable one. Homeopaths think that their medicines contain information, held by the water/alcohol mixture in which they are made. From the chemical view point homeopathic medicines consist of water, alcohol and lactose (from the pills).
But think of a floppy disk or video tape: chemically it consists of vinyl and ferric oxide), but it can store large amount of information. This information is stored in physical form (the aligment of the dipoles of ferric oxide), which cannot be detected by chemical analysis. There is now growing evidence that the process of preparation alter the structure of the water and this modification of the water carries information.