Externship: done during school. Involves spending time with one or more vets in sort of a participatory shadowing. It's a part of 4th (and maybe 3rd) year clinical rotations.
Internship: advanced clinical training after graduation. You're a doctor but under the supervision of more senior doctors to at least some extent. You do get paid, though not very much.
An externship is taking during your time at vet school - usually during the 3rd or 4th year. It is generally 2-6 weeks in duration but that varies with the site. Essentially, you go someplace else to learn about vet med for a little while. It is required at my school for the 4th year - you have to have at least 4 weeks taken during that time that are graded. They are also called preceptorships.
An internship is a year long and takes place after you receive your DVM. It is intensive and is usually in a specific area like small or large animal.