A great article about preparing for an internship and how to get the most out of an internship: https://www.gamasutra.com/blogs/EthanLevy/20130903/199493/7_lessons_I_wish_I_learned_before_starting_my_first_game_internship.php
Some current internship opportunities:
Microsoft: primarily software engineer positions
A niche opportunity: Software development internship with the St. Louis Cardinals
Interaction Gaming (in Ann Arbor, MI): QA and Development (looking for a student pursuing CS)
Twitch: Engineer internship
PerBlue (in Madison, WI) has internships and full time positions open
Immersed Games (Buffalo, NY) has programming, UI and Writer internships and Game Art jobs open right now. http://www.immersedgames.com/become-an-immtern/
ArenaNet (Bellevue, WA) has an animation internship and a creature artist internship
Zynga https://www.zynga.com/job-listing-category/students-new-grads-interns/
HTC: 3D artist generalist internship
TWO BIG COMPANIES WITH A LARGE NUMBER OF INTERNSHIPS:
Sony Playstation has a huge amount of internships
Activision has some and will continue to have more