I agree with what Tom and ambershee said.
At the studio I work at:
- We DO take on a very small number of interns each year. NOTE this is for university placement years, not for short term school "work experience".
- It's done in partnership with universities close to our office. The best N students are chosen from CS or games courses.
- AFAIK we don't take on design interns, only programming and art.
- The most successful candidates have a real passion about games and making games; they do a lot of stuff in their own time.
As Tom says, expect to apply to a lot of companies to get an internship (all of them if you have the time!). Expect to apply to even more to get a placement as a designer - there is no shortage of good ideas (and more people queuing up with good ideas) in the games industry - what is needed is people who can make those ideas a reality - that's why people who get the pure design jobs tend to have experience. You may find intern roles as a "level builder", but quite often that's considered a branch of environment art so unless you have a qualification in say architecture or town planning, or are good at art, that might be out.
The competition for intern and entry level games positions is fierce - think about it, everyone in the world currently doing a games course at university or college wants those positions, so do a large number of the people on this board and similar. So as well as a lot of luck, you need to stand out from the 1000s of people you're in competition with.
How many games have you made yourself? Do you enter competitions like Ludum Dare? (etc etc) How many mods/levels have you made for existing games? Have you learnt to do any programming and/or art? Have you showcased your games (and other creations) much on forums such as this one? Have you covered a broad range of game genres in the stuff you've made (when it's a real 'job' you have to produce good work even for genres and IPs you hate)?
The people who are getting those intern roles are doing all of those things and more, if you aren't you should be! Those things are also major points in your favour for when you apply for non-intern entry level roles.
If you have stuff to show, then include links when you're contacting companies. Local games industry networking events can be a mixed bag - they're a good place to meet student, indie and hobbyist developers who you can collaborate with on more games. They can be a good place to get advice of people in the industry. If you have a good portfolio of games (and related things) you've made, it can be a good place to show people (to get feedback, and to ask if companies have any intern places available).
Game development conferences are good for similar reasons if you can afford to go and have a higher proportion of companies and professional developers.
As a group, the company I work for now also has a graduate scheme that might be of interest to you: https://www.ubisoftgroup.com/en-us/careers/graduateprogram/