I won't go into deep details on Homeopathy (how it was invented, where and by who, etc.) but look, briefly, at what it is and claims to be.
Homeopathy is based on the idea of "like cures like". For instance, if eating hot peppers makes you sweat then hot peppers should cure excessive sweating. Homeopathy suggests taking a small amount of the "active ingredient" (ie. hot pepper) and diluting it to 10% and then taking the resulting solution and diluting it to 10% again and then taking the resulting solution... sometimes hundreds and hundreds of times. The solution, by homeopathic claims, becomes more "potent" with each dilution. They suggest that water "has a memory" of the original ingredient. A drop of that "water" is often then put onto a sugar pill (pillule) and sold as a homeopathic "cure" or "treatment" for a vast number of symptoms/diseases.
Some homeopaths might claim there is more to it than that but there truly isn't.
The initial premise is "like cures like" and, unfortunately, there is no evidence to support this. From a critical thought standpoint, why should someone even suppose this to be the case? The logic does not follow and it goes against much of what we know from modern science.
Assuming that the initial premise was even plausible, the next stage of diluting it to the point where the likelihood of finding a single molecule of the original "ingredient" in the resulting solution is so tiny that it almost certainly isn't there, has to raise questions.
To solve that logical problem, homeopaths often claim that water has a memory. "Easy", they say, the water "remembers". If the water has a memory, how does it know what to remember? Does it remember any of the soap particles or dust it might have come in contact with at some point in just this process? What about from before that process - when it was in a sewage treatment plant or passed through the bodies of animals?
Sound silly? There's more. Homeopaths claim that the more diluted the substance is, the more effective the treatment is. If this is true, you could overdose by simply taking too little - simply by drinking too much water when you swallow the pillule.
A 30C dilution (Hahnemann, the guy who 'made up' Homeopathy, often advocated this dilution) would, on average, require giving two billion doses per second to six billion people for 4 billion years to deliver a single molecule of the original material to any patient. (See ScienceBasedPharmacy)
And that's just plain silly.
Other reading on Homeopathy:
Edzard Ernst and Michael Baum have an excellent article in the Journal of American Medicine, Should we maintain an open mind about Homeopathy?
Recently (on January 30, 2010), to show the absurdity of Homeopathy, skeptics swallowed entire bottles of homeopathic remedies - with no ill effects (maybe a rise in blood sugar from the sugar pills).