While hydrogen peroxide can be useful for treating wounds and has some dental benefits, most of the claims that hydrogen peroxide can cure various diseases are not backed by science. Diseases like cancer and HIV, in particular, can't be cured by any hydrogen peroxide therapy.
Hydrogen peroxide is famous for its bubbling when applied to certain surfaces and chemicals. Hydrogen peroxide forms oxygen and water when it interacts with certain chemicals, including many that collect in cuts. Hydrogen peroxide has weak antiseptic properties, but its bubbling action makes it an effective way to clean wounds. However, it also damages healthy tissue a bit as well, so it should be used sparingly.
Dentists also use hydrogen peroxide for a number of therapies, and it can be used safely for mouth rinses. Many tooth-whitening formulas are based on hydrogen peroxide, although homemade formulas are potentially dangerous. However, unless directed by a dentist, most people see little benefit from hydrogen peroxide if their dental hygiene is sufficient.
A number of people make bold claims of hydrogen peroxide curing a wide range of diseases, but experts emphasize that these claims are unsupported and sometimes refuted by evidence. In addition, there's little reason to believe that hydrogen peroxide can cure cancer or other diseases known for being difficult to treat.