What is homeopathy?

Homeopathy is a sub-system of natural health care in which extreme dilutions of substances from nature are used to stimulate a healing response. The basic principle of homeopathy is that substances that elicit a particular symptom picture in their physical form (for example onion causing itchy watery eyes and a runny nose with discharge that burns) can be used in a diluted form to stimulate the healing of the same set of symptoms.
Using this principle, a homeopath working with an individual that suffers from allergies (with the specific symptoms of watery itchy eyes and a runny nose with discharge that burns) might select homeopathic onion.
Homeopathic pharmaceutical companies prepare homeopathic dilutions according to FDA guidelines. Once completed, the diluted substances are called “remedies”. 
Most remedies are derived from plants (such as Belladonna or Pulsatilla), and minerals (such as Phosphorous and Silica). A smaller portion are derived from animal substances (such as dog's milk or dolphin's milk). Remedy symptom pictures are derived from research studies called “provings”. Once completed, proving information is recorded in homeopathic materia medicas (large reference books), and homeopathic material medica software.
The role of a professional homeopath is to understand the symptom pictures of a large number of remedies, and then to be able to select the remedy that most closely matches the symptom picture of the patient. (This is called the similimum.) When this match is achieved – that is, the symptom picture of the remedy matches the symptom picture of the individual suffering from a disease pattern – the individual's innate healing ability is activated and healing occurs.

How do homeopathic professionals use homeopathy treat cancer? 

There are several different treatment philosophies and approaches for using homeopathy to treat individuals with cancer. Often times practitioners that use homeopathy in the treatment of cancer use a combination of these approaches. 
One approach is to use homeopathic remedies to target the tumors themselves. In this case, the homeopath selects remedies which match the symptom picture of the tumor itself (e.g. Conium Maculatum for hard immovable tumors that develop slowly).    Homeopaths using this approach might also consider other symptoms (such as the individual's food cravings, disposition, etc), but their primary focus is to target the tumor and reverse its growth. Some physician homeopaths also give remedies at the tumor site itself (in the form of an injection) to more aggressively stimulate a response.
Another approach is to use homeopathic remedies to assist in healing the patient's eliminative channels (kidneys, urinary tract, lymphatic system, liver, etc), and strengthen cell detoxification. In this case, the homeopath may use drainage remedies. These are low potency combination remedies that are used to target specific systems, or detoxify particular substances (e.g. heavy metals, etc). Homeopaths using this approach select the drainage remedy based on an analysis of the case – or they may use an electro-dermal diagnostic tool, such as the Quantum machine.
Still another approach is to use homeopathy to address the overall constitution of the patient. In this case (called the classical approach), the homeopath does a complete interview of the patient's mental, emotional, and physical symptoms and then selects the best match accordingly. Often the best selected constitutional remedy will directly affect the tumor. By assisting the body in addressing and resolving the energy that underlies the tumor, it can result in complete elimination of the tumor. The best selected constitutional remedy can also support drainage and detoxification, although this is not the primary focus of a classical prescriber.

How effective is homeopathy in the treatment of cancer?

In the US, most alternative health care professionals use homeopathy in combination with other approaches (nutritional and immune system support, diet and detoxification programs, etc), so it's not typically considered as a primary or stand-alone therapy for cancer. However, when used correctly, and in combination with necessary therapies, it can be an amazingly powerful tool for empowering someone to heal themselves of cancer.  There are many cases in homeopathic literature (from other countries where homeopathy is more commonly used for cancer) in which the correct homeopathic recommendation resulted in the resolution of tumors, reversal of the cancer process, and recovery by the patient.
Also, homeopathy has an excellent track record in the treatment of animals with cancer and there is much anecdotal evidence that certain remedies (when correctly selected) are effective in reversing the cancer process.
Much of the success with homeopathy, however, depends on the skill level of the practitioner and the resources of the patient to heal on all levels.