The Claims
Homeopathy was created in the 18th century (late 1700s) by Samuel Hahnemann. Perhaps back then it may have seemed plausible. After all they did not have the advancements in medicine that we have today which have enabled us to double our life expectancy in the past 100 years. Today however any reasonably intelligent person can see this outdated system for what it is, many people however simply do not know what homeopathy actually entails.
Does how is homeopathy supposed to work? The basic idea is that an ill person can be cured by a substance that produces similar symptoms in a healthy person. So for example, a skin rash could be treated by a solution of poison ivy as undiluted poison ivy can cause a skin rash. This of course of nonsense - unlike a vaccine in which a diminished form a virus can be introduced to the body to induce the immune system there is no connection between different viruses and diseases that happen to cause the same symptoms.
If this wasn't strange enough, the next claim made by homeopathy is that the more you dilute the active ingredient the more powerful a remedy it becomes. To this end, modern remedies, made to the recommendation of Hahnemann himself, are so diluted that they don't contain any molecules of the active ingredient at all - they are just solutions of water.
How then do they claim that it works, given most producers will fully admit they are just selling water solution? They claim that the water can remember what it has been in contact with. Yes, we're being serious - water has a memory. Interesting however it is unable to remember everything else it has been in contact with such as salt (from the sea, quite a lot of water there), urine (we all pass water through our bodies) or the chemicals used to sanitise water before it is delivered to us by our water companies.
In fact, the laws of probability show that for every glass of water you drink, at least one molecule has passed through the bladder of Oliver Cromwell. There is nothing special about Cromwell, he has just been dead long enough for the odds to stack up. The chances that a molecule has been in contact with the active ingredient is far smaller - yet it is supposed to "remember" this, even though water clearly doesn't have memory, and not the inner workings of Mr. Cromwell.