Homeopathy is an alternative to medicine which was invented in the mid 1700s by Samuel Hahnemann who thought up the idea of "similia similibus curentur", or "Like cures like". What this means is that he believed if you had a symptom, giving you something that causes that symptom would cure it. For example, giving you poison ivy to cure an itchy rash. Not only that, but Samuel decided that diluting the substance to the point where there wasn't actually any of it in the water anymore made it even more potent.
Mad Art Lab has covered an excellent analogy for the dilutions of homeopathy. People seem to believe that higher numbers on a homeopathic product indicate higher concentrations, but the opposite is actually true. 1X would be one part in ten of the active ingredient. 1C would be one part in 100, and is the starting point for homeopathy. You would then dilute it to 2C, then dilute some of the 2C to 3C (1:1,000,000), until you get to the numbers commonly seen on homeopathic products. 4C is practically inert; a 4C dilution of arsenic in your drinking water would be completely safe to drink. At 12C there is only a 50% chance that a beaker of the homeopathic product has even one molecule of the original ingredient in it. Consider that many homeopathic products have an even greater dilution than THAT and it's easy to see why homeopathy is complete quackery, and considered by many to be unethical. Studies show these products to be no more effective than a placebo. You are literally paying for water.
But, you may protest, I gave my baby a homeopathic teething remedy and he stopped crying! How can a baby be affected by the placebo effect? I'm glad you asked. People commonly mis-understand the placebo effect, and believe that you must know the desired effect of the treatment in order for it to work (for example, the baby would have to know that he was being given a pain reliever in order for a placebo to relieve pain). This isn't quite accurate. In the case of babies (and pets, actually) the placebo effect can also work on the expectations of those observing the subject (mom feels she's done something and is less anxious, and baby picks up on this). Additionally, intervention itself is likely to cause a reaction. Holding the baby to administer the medicine might be soothing in and of itself. Or just interacting with the baby can calm them; children often are soothed by mommy kissing their boo-boo better, but nobody would market kisses as an effective medicine. Teddy bears calm children, but we don't sell them as treatments. There are plenty of studies showing at least a semblance of placebo effect in young children (though it should be noted that it does not affect neonates).
On a slight tangent, I worked in retail pet stores for several years and you can often find homeopathic products for dogs and cats. One that I find particularly abhorrent is a homeopathic de-wormer. This has a potential for real harm, because a well-meaning pet parent might purchase it and believe their pet has been rid of parasites, when in fact the pet will continue to harbor them and may fall seriously ill. This is less likely to happen for children since the most common homeopathic product for babies is for teething, and as discussed above they will derive a placebo effect from it (you can't "placebo effect" away an infestation of roundworms, though).
For all of these reasons, from complete lack of active ingredients to complete lack of real effect, I am completely against the use of homeopathic products and wish they were not available for sale. I hope this has clarified the realities of homeopathy as an alternative to medicine!
Sources:
Placebo effect in neonates
Mad Art Lab: Diluting Felicia Day
Wikipedia (and sources listed)
