Few. Infections that are caused by bacteria (including gonorrhea and chlamydia) can be cured with antibiotics if they are caught early, while infections caused by viruses (like herpes, hiv, or hepatitis c) can often be controlled somewhat with medications but can't be completely eliminated from the body. Finding any kind of infection early is best, so see your healthcare provider if you are concerned!
STDs. Some are chronic but many are treatable and can be cured.