A systemic infection is so named because the pathogen that causes it, and often the symptoms that it causes, are spread throughout the systems of the body, instead of being localized in one area -- as they are in a local infection. Systemic infections are not necessarily more severe than local infections, they just affect a larger proportion of the body.
Not all systemic diseases are infections. For example diabetes causes symptoms and changes throughout the body, and it is often an autoimmune disorder. Cardiovascular disease, which is affected by behavioral factors, genetics, and the natural processes of aging, is also a systemic disease.
Several STDs are either always systemic infections or can become systemic infection. HIV, for example, is a disease of the whole body. Gonorrhea on the other hand, is usually a local bacterial infection, but it can become disseminated under certain circumstances.
Interestingly, although chlamydia may seem like an obvious candidate to cause systemic infections, since it can ascend to the uterus and affect a variety of sites including the eyes and the rectum, the specific type of chlamydia that causes genital infections is not generally thought to cause systemic infections. Other chlamydia types, however, may do so. For example, the systemic disease lymphogranuloma venereum is caused by a type of chlamydia, even though the infection behaves more similarly to syphilis.