"Homeopath" may sound like a cool new portmanteau to describe gay serial killers, but homeopathy is actually a very old and widely disproven form of alternative medicine. Don't feel too disappointed about the lack of serial killing; encouraging people to shun medicine that works in favour of alternative medicine that doesn't can occasionally result in the patient dying of a disease which is curable, or at least treatable, such as asthma, food poisoning and even fatigue. It's also a great example of what can happen if you try to develop a system to solve a problem (in this case, ill health) without using logic or science. There are two main ideas to homeopathy, both of which are completely wrong.
The first is that "like cures like." Homeopathy teaches that if one thing has a certain effect on you, and another thing has a similar effect on you, then you can use one of these things to cure the other. For example, say that staying up all night makes you tired, and that listening to your aunt's tales of her childhood also makes you tired. Someone with this faulty sense of logic might jump to the hasty conclusion that listening to a little bit of one of your aunt's tales might make you immune from feeling tired after staying up all night.
Obviously, it doesn't work that way. Just because two things might have similar properties or cause similar effects doesn't make them interchangeable or interlinked in any way. The closest this line of reasoning comes to the truth is that it's superficially similar to a vaccination, which actually does consist of a weakened, inactivated or dead pathogen that would otherwise make you ill. In the case of the vaccination, however, it works because it's a safe way of giving you exactly the same thing that makes you ill, not something a bit like it. This allows your body to produce antibodies that learn how to fight off the disease injected into you because, after billions of years of evolution, you've developed a totally awesome immune system.
The second completely wrong main idea of homeopathy is that watering something down more makes it more potent. Again, this may look superficially similar to a vaccination, but the watering down goes even further than that. The dose of the unrelated item that causes vaguely similar symptoms to your own is watered down so much that there would only be one drop of the substance in all the water on Earth, the Sun, Venus, and all the empty space between them, if all that space were nothing but a big ocean of pure water. This is, of course, impossible. "Stronger" (actually weaker) doses have only one drop of the substance in question in amongst more water than there is tangible stuff in the entire universe. These doses were, naturally, dreamed up before we knew what a molecule was, and before we knew how many there are in the universe.
As you may have noticed while idly forgetting to take some real medicine, taking less of it does not make its effects more pronounced.
Advocates of homeopathy claim it works and that scientists have either ignored it, or proved it works, or want to keep it hush-hush and so on. There have been hundreds of experiments performed on homeopathy, and all the carefully executed ones show that its effect is the same as a placebo. So although it can cure anything that any other placebo can cure (such as depression and gastric ulcers), it's clearly not the homeopathy itself doing it, but the placebo effect. The placebo effect is a fascinating thing, and I'd recommend reading about it just because it's so counter intuitive. However, it doesn't make homeopathy any better than eating sugar pills.
On the subject of sugar, it's worth noting that some homeopaths don't even directly give patients the water that used to have something in it that causes similar symptoms to their own. Instead, they pour the water over sugar, wait for the sugar to dry out, then give them that. No wonder homeopathic treatments fare no better than sugar pills -- they are sugar pills.
Naturally, some people feel like homeopathy, along with any other alternative medicine, has worked well for them in the past. This is sometimes caused by the placebo effect, but sometimes it's simply a case that they were going to get better then anyway. Most ailments only last a few days, and no matter what you do in that time, you can easily convince yourself that the thing you did caused the symptoms to disappear. Only with large groups of people, half doing the same thing as you, and half not, can we weed out the background noise of coincidence and observe whether the thing being done differently has actually caused the symptoms to disappear or not. This is exactly what a scientific experiment does, and as previously mentioned, all carefully performed scientific experiments have proved homeopathy not to work.