Homeopathy involves two steps: one nonsense and the other complete nonsense.
Homeopathy involves treating 'like with like'—in other words, treating a disease that manifests itself through certain symptoms with substances that induce those symptoms. In this light, homeopathy claims some similarity to the immunological principles we are familiar with as immunisation, vaccination or inoculation. BUT—in the case of immunology the 'like' really is with 'like'—thus the induced symptoms, albeit milder, are real as immunisation involves introducing into the body an antigen that is either the real thing, albeit attenuated by heat or chemical treatment, or an analogue of the real thing eg cowpox (which stimulates the body to produce antibodies against smallpox) instead of smallpox (which could be fatal). However, in homeopathy, the substances chosen have no biological immunological actions, they merely purport to mimic the symptoms of the disease. This is nonsense as there is—unlike in immunology—no theoretical or demonstrable biological basis for the relationship between the disease and the purported causative agent. This is like saying that migraines cause headaches but so does a blow with a hammer—so hit the migraine sufferer with a hammer and the headache will improve.
It gets worse. Patently, homeopaths are not stupid (they just expect us to be) as they do not administer any harmful substances to people. Instead, they dilute the active ingredient out of their treatments by a series of dilutions (called serial dilutions) and also some 'hocus-pocus' knocking of the containers of the solutions a certain number of times until there's nothing left in the solution. It is customary to argue this point using Avogadro's number; but this is pointless (although I've used it in the past) as homeopaths admit that the serial dilutions lead merely to water. However, this is not just any water; it is water containing a 'memory' of the substances that were once being diluted in it. This is the complete nonsense of homeopathy—there is no trace of anything, real or imagined; there is absolutely nothing there. Talk of 'memory' in the water is based on the ability of the substance to arrange the water molecules around it, thereby shaping the water and—if this were possible—that memory is the active ingredient in homeopathic remedies. Water is highly dynamic—that is why at the temperatures we encounter it, it is a liquid. Water molecules interact—and water remains liquid water—by means of hydrogen bonds; these are relatively weak interactions (compared with chemical covalent bonds) existing due to the uneven distribution of electrons in the bonds between hydrogen and oxygen. Weak though they are they create the surface tension of water which is visible when water forms drops or you lightly place an object on the surface of water—the surface 'bends' up to a point. To break the hydrogen bonds in water a temperature of 100 degrees centigrade is required. So, how does a substance leave a trace in water that is dynamic—hydrogen bonds are continually forming and reforming between water molecules; the answer is, simply, it can't.
Does homeopathy work?
The question of whether or not homeopathy works is not a sensible question; the proper question is 'can homeopathy work?' and, of course, the answer is 'no'. Homeopathy cannot work as there is nothing in it that can work. But people get better using homeopathy—don't they? Yes they do but people also get better if they do not seek treatment or even if they take the wrong treatment; nature takes its course and this applies not only to homeopathy but to a great many 'conventional' treatments. People think it works for two reasons: first, they were getting better and they used homeopathy—this is reverse causation; second, people sought treatment from a homeopath and they got better because they believed in the treatment and enjoyed the experience—the placebo effect. The placebo effect is not to be ignored; it is very powerful and accounts for a great deal of effect in conventional medicine. However, the ultimate question is 'does homeopathy work better than a placebo? and the answer here is 'no'. There are isolated reports—some in prestigious journals—of effective homeopathy; but there are also reports of no effect...or worse. Isolated successes do not constitute proper evidence and the evidence—such as it is—needs to be studied over time, combined and evaluated to eliminate bias, poor designs and any other explanations. One major explanation for isolated successes with homeopathy is regression towards the mean and this phenomenon will occur with any treatment, however unconventional or unlikely. Most of the time the effects of any treatment are not that spectacular but sometimes a treatment will appear to work very well in a well designed study; however, wait long enough and another well designed study will throw up the opposite result. It is in combining these studies that evidence is accumulated and, sadly for homeopathy, the evidence for an effect is not there.
Is homeopathy amenable to scientific study?
Homeopaths say 'no' because they claim that there is a lot more to administering a homeopathic treatment than handing over a solution (water!)—that the whole assessment process and interaction with the patient is important. This is a smokescreen—the same applies to a GP administering antibiotics (or it should!). On the contrary, homeopathy must be the simplest and safest therapy to investigate—provided that life-threatening diseases are not being investigated: simply take a patient through the whole routine of assessment and interaction and then hand them a purported homeopathic remedy (water!) or a remedy that is known to be water but which has not been through a series of dilutions and which has never contained the purported active substance—easy!