Internship: A temporary employment at a company or office (such as a summer internship at a firm). The tasks vary, but are usually tedious and simple work that does not require much knowledge of the work/industry. Internships are common for college students and may be paid or unpaid. The goal is usually so the intern may test to see if they like the company, the work, and to form relationships with their co-workers in the hopes that they might be offered a full-time job later.
Apprenticeship: A period of work where the apprentice learns on the job, while working alongside an expert until he/she has gained enough experience to work alone. Apprenticeships are usually for trades - such as for electricians, plumbers, chefs, carpenters etc., - not for office work.
On-the-job training: A short period of time when one is trained while working a long-term job. May be in an office or not (such as bus drivers).