In most cases, you are directly enrolled in an overseas university, taking courses and sitting for exams with regularly enrolled students.
Exchanges can take place at any level, from sophomore to graduate.
An exchange can last a full academic year or one semester. With prior planning, credit earned abroad will be applied toward your degree. Also, most academic departments will grant credit for study abroad. Early approval from your advisor regarding credit transfer will reduce later worries. Many opportunities are available in English-speaking countries.