This is Soursop, otherwise known as Graviola fruit (Annona muricata), or guyabano.
It has been used for many years as an herbal medicine. It is native to Central and South America as well as the Caribbean and is also grown in parts of Asia.
A number of sites have articles all detailing the same information with slight variances.
For example: http://aoand.com/profiles/blogs/graviola-tree-10-000-times-stronger-killer-of-cancer-than-chemo and http://www.jamaicans.com/blog/?p=347
The claims commonly ascribed to this fruit are (copied from claim supporter site):
*Attacks cancer safely and effectively with an all-natural therapy that does not cause extreme nausea, weight loss and hair loss
* Protect your immune system and avoid deadly infections
* Feel stronger and healthier throughout the course of the treatment
* Boost your energy and improve your outlook on life
* Effectively target and kill malignant cells in 12 types of cancer, including colon, breast, prostate, lung and pancreatic cancer
* The tree compounds proved to be up to 10,000 times stronger in slowing the growth of cancer cells than Adriamycin, a commonly used chemotherapeutic drug!
* What’s more, unlike chemotherapy, the compound extracted from the Graviola tree selectively hunts down and kills only cancer cells. It does not harm healthy cells!
After a fairly extensive search I only found 1, peer-reviewed, scientific article describing the benefits of Graviola extract (which is not the same as raw fruit). You can read it the abstract here: http://www.ncbi.nlm.nih.gov/pubmed/21767082
In the article it says that the extract "significantly downregulated EGFR gene expression and inhibited the growth of BC cells and xenografts." EGRF is a gene commonly found in breast cancer. The dosage of the extract was 200mg/kg which in an average human female equates to 6.4 grams per day for 5 weeks. However, the study was only done in mice and so the therapeutic equivalent could be much higher.
Further claims of feeling stronger, energy boosts, improved outlook on life are all things that a simple can of Coke can provide or even being outside in the sunlight. They're also claims that are hard to show definitively came from the fruit since they're all subjective and can be caused by any number of everyday things.
The dubious claim that it is 10,000 times more effective than chemotherapy is one hard to substantiate. Even if it were 10,000 times more effective than Adriamycin (which is actually an anti-biotic used in cancer treatment) it is still only more effective than a single drug which is used in a limited number of cancers. Many people have taken the information to mean that it is more effective than all chemo agents against many cancers.
It is also purported that since Graviola comes from a natural source it is better for you than chemotherapy which is an evil man-made poison. However, Adriamycin (doxorubicin) and many other chemo agents and other drugs are derived from natural sources. Adriamycin comes from a bacteria, Taxol and Taxotere both come from the bark of the Taxus brevifolia tree, Thiopurine drugs are derived from Purine, an organic compound found in our DNA, the entire Anthracyclines and Camptothecins classes of drug come from natural sources like plants and fungi.
Another problem with the "Soursop cure" is that there are no verifiable third-party sources to be found in any of the pro-Soursop articles I have read. While they list groups like the Catholic University of South Korea and the Journal of Natural Products, other than the single article I mentioned above, none of these sources can be found. Even Wikipedia requires that facts or claims be backed up with 3rd-party sources.
One thing we do know is that while Soursop may have some benefits against breast cancer in mice it is also dangerous. The fruit contains high levels of annonacin, which in large doses (3-8mg/day) can lead to Parkinson's Disease. It can also cause general cell death by blocking the cell's ability to produce ATP which is basically the bodies source of cellular energy.
Proponents claim that there is some vast conspiracy to keep the "truth" about Soursop away from the public eye because, since it is a natural product, it cannot be patented and so profits are low. However, there is nothing to prevent you, a doctor, chemist, hospital, university or other group from conducting their own scientific studies of this fruit and publishing their findings. The theory that profits can't be made by selling healthy and natural products is absurd. GNC brings in over $2 billion a year and a study by the Natural Products Foundation showed that the dietary supplement industry alone has an economic impact of $60 billion/year.
The bottom line: Soursop contains some compounds beneficial to breast cancer, however eating or drinking the raw fruit may not yield any benefits at all. It is also potentially dangerous to do so. I can not find any scientific information to support the claim that it is 10,000 times more effective than chemotherapy or is greatly more helpful in other cancers than current treatments. The idea that this is a cure-all or that direct intake will slow or cure cancer is completely false.
Sources:
1. Chemo Natural Sources http://www.uesc.br/cursos/pos_graduacao/mestrado/animal/bibliografias/roueda/artigo3.pdf and http://www.fmh.org/body.cfm?id=900
2. Causes Parkinson's http://www.ncbi.nlm.nih.gov/pubmed/14675150
3. GNC Revenues http://biz.yahoo.com/ic/15/15942.html
4. NPF Report http://www.naturalproductsinfo.org/index.php?src=news&srctype=detail&category=DSIB%20Releases&refno=181&view=DSIB_Releases_Detail
Also:
1. http://www.hoaxorfact.com/Health/graviola-tree-10000-times-stronger-killer-of-cancer-than-chemo.html
2. http://en.wikipedia.org/wiki/Soursop
Disclaimer: I used to work as a contractor with the US Veterans Department in one of their pharmacies so I may be a part of the grand scheme to block you (a free individual) from learning about a tree (that can be found world over) or from using the scientific method to discover its uses.