Homeopathy consists of medicines which are diagnosed based on a very specific symptom picture matched to substances (plant, mineral, etc.) which cause the same condition (physically/mentally/emotionally) in a healthy person. These medicines are often given in amounts so small that they no longer contain a molecule of the original substance but instead seem to leave some sort of energetic signature in the medium (usually water or milk sugar pellets).
It is my understanding that homeopathic has enough research to prove unequivocally that it has significant effects beyond placebo. However, many clinically trained homeopaths will agree that good research is in short supply. As with many alternative medicines, there is a great need for more research funding and research methodology that reflects the way the medicine is actually prescribed and used. Homeopathy has been subject to various studies which either use medicines prepared incorrectly or attempt to prescribe them based solely on conventional diagnostic criteria. Such studies are doomed to failure from the beginning. Despite this, research has been conducted, the best of which is compiled in the book: Impossible Cure by Amy Lansky. The book gets into homeopathy as a whole from various angles and is recommended reading as a proper introduction to homeopathy.
Here are two of those studies:
1997 meta-analysis resulting in an odds ratio of 2.45 in favor of homeopathy: http://www.ncbi.nlm.nih.gov/pubmed/9310601
Acute Diarrhea study in Nicaraguan Children: