系
is used exclusively for a department in a university, and you should not use any other word for it. (Of course, Faculty of Engineering/College of Engineering are translated into 学院 like 工程学院, but that's an upper-level organisation of a department)
When referring to a department in a hospital, use 科
. For instance, department of radiology/internal medicine/obstetrics and gynaecology are called 放射科/内科/妇产科, respectively.
When referring to a government division or office, the names could vary among 局, 处, 厅, 部, etc
depending on the level of that government division. For instance, Department of Personnel of the central Chinese government is called 人事部, of a provincial government 人事厅, of a city government 人事局, of a public university (which is sort of part of the government system) 人事处. This is the naming system in mainland China. The names are different in Hong Kong and Taiwan.
When referring to commercial organisations, it is typically 部门. A company's department of human resources would be called 人力资源部.
The established name of a department usually contains a single character 部 (department of procurement 采购部), while 部门 would be used for generic references such as IT部门 (IT department).