My company recently had an opening for a junior game artist position and I was really surprised by the amount of interest it received: over 250 applications in a bit more than a week, even though we're not exactly a high profile game developer. As I looked at the applications, I noticed the same kind of oversights kept showing up and often made it difficult to consider those applicants for the next step.
I was a junior artist once but in the past few years I've had the opportunity to hire over 15 2D and 3D artists. Here's the kind of advice I would have liked to receive when I was fresh out of school and desperate to get a foot in the door. Some tips might seem obvious but I still decided to include them because I kept seeing the same faux pas again and again. Your context might obviously be different so feel free to discard some of points below, but overall I hope they will help you land your dream job!
The portfolio is by far the most important piece of your application, the thing the company will check first because it's the fastest way to assess if you're qualified for the job, but somehow it seems a good 5 to 10% of applicants don't provide an easy way to actually see any of their work. Without a portfolio there's a good chance your application will be rejected right away.
You want to work on games so show how your skills can be applied to games in a very concrete way. Make it easy for the company and present art that relates to the work they might ask you to do once you're in: characters, menus or even mockups for imaginary games, similar to what you would see on the screenshots on the App Store or Steam. Game Jams can be a fun way to enrich your portfolio with such material.
Graduates often have a wide portfolio (different styles or types of work) but the company will also want it to be deep enough (several examples in the same style). It's hard to judge somebody's skills and consistency on a single asset: rather provide a small set of items or characters all in the same style, as if it was for a game. Similarly, if your portfolio only contains the few projects you were required to do during your studies, you're going to be at a disadvantage against other applicants who can also present personal projects that are often more valuable to assess the skills of an applicant.
You want to show you can manage the creative process from start to finish, so show primarily finished artworks that look like they are ready to be used for the production of a game. Too many unfinished pieces in your portfolio may make you look like you lack focus or give up easily.
Present your strongest images first. What ‘strongest' means might change depending on the position or the company you're applying at. Chronological order is often not ideal and that's why having your portfolio on Facebook or Instagram is not recommended. If you have multiple skills, split your portfolio into different sections, like 2D, 3D or UI so the company can focus on exactly what they are after.
If all your work is marked as being from a few years ago, the company is going to wonder what you've been doing lately. To avoid this you might even consider not to date your pictures so that they look evergreen.
Be wary of using tests you've done for other companies in your portfolio. If they're recognizable (some studios ask for the exact same test for years), it's also a way of saying "I was turned down by those guys" which might make the company you're applying at reconsider.
Because apparently it's not obvious to everyone, your portfolio should contain professional work only. It shouldn't be an Instagram account where your artwork is mixed up with selfies and photos from your holidays. Similarly be wary of applying with any art that could be considered NSFW as that might not always be well received.
A cover letter explaining your situation is good, especially if your profile is unusual for the job, but keep it short, a few lines is usually enough. Without one the company might be left wondering what position exactly you're applying for, whether you're willing to relocate, etc. It's also helpful to know what decided you to apply in the first place, as long as you're relatively honest - excessive flattery is easy to spot often frowned upon.
If you're foreign-based but are entitled to work in the country you're applying for, mention it. It's often a big deal for the company as it might not be able to offer visas for junior positions. Sponsoring a visa takes time and money, two things that are scarce at a small studio. If you have a passport from a different country that the one you currently live in or have several passports, mention it.
For a junior position, your resume should fit on a single page. There's no need to detail your past work experiences if they don't have a clear connection with game development. You can always talk about this at the interview stage if needed.
Make it easy for the company to find your portfolio by including a link on your PDF resume and make sure it's clickable - or at the very least selectable. Nobody wants to have to type a long URL in the browser address bar when they have many applications to go through every day.
Fancy-looking resumes are cool but they should still be easy to read. If in doubt, better play it safe, you won't get turned down for having a standard-looking resume. Correct grammar and punctuation never hurt either, so ask someone to double-check your resume if you're not sure!
If you apply for a job abroad, do some research first of what is expected to be (or not expected to be) on the resume in that specific country, as it varies a lot. In some countries including a photo of yourself on your resume is normal, in some others it's very unusual.
If you're from a non-English-speaking country, applying to a company abroad with a resume that has not been fully translated into English or the local language doesn't show a great deal of motivation for relocating.
Talent and skills take time to acquire, learning a new tool not so much. If you're a good fit, nobody in their right mind is going to turn you down because you use SAI instead of Photoshop, especially at a small company.
If you're applying at a small studio, show that you can do anything relating to game art competently (not just concept art, but also 3D assets, sprites, full game mockups with UI, logos, etc.). If you're applying at a big studio, show you're extremely talented at a specific skill (ideally the one the job is about!) but also competent at other tasks.
Being passionate about games is a plus for working in the games industry, but that's not what you'll get paid for. Don't let your passion as a gamer get in the way of becoming a talented professional. You probably won't always have the chance to work on the exact type of games you love playing, but don't let that impact the quality of your output. Instead, find a part of the job you can be passionate about even if you're not the target for the end product. You might dislike racing games but you could still enjoy working on one.
Read the job ad carefully, research the company and apply only if you honestly consider you're a good match. Look at your portfolio - does it objectively look like the kind of art the company you're applying to would make a game with? Ask a good friend with knowledge of the industry to give you his honest opinion if needed. Competition in the field is high so you won't get hired by default. Avoid the scattergun approach and be a sniper instead, carefully crafting applications for positions you really care about.
Research the company you are applying to and if possible, include some art in the style of their games in your portfolio. If a company only produce cute games and you apply with a portfolio full of ultra-detailed monsters, it might not be the best approach. Maybe you can also do cute but if it's not in your portfolio the company will assume you don't enjoy doing it. Similarly, if you're a 3D artist who applies for a 2D position (and vice versa), you can be pretty sure you won't be considered as the aptitudes required are quite different. If your style or skills are very specific, target companies that already make use of those.