The law enforcement agency responsible for carrying out court orders or summons or operating the county jail is known as?

Different states use different agencies to carry out various legal functions. In Florida the Sheriff's Department supervises the county jail. Most states follow that pattern. In some a different organization is in charge. In Florida, it depends on the court order as to who carries it out. Sometimes it is a deputy sheriff. Sometimes it is a payroll officer. Sometimes it is someone else. Your state may have its own rules.