Private companies are privately held without the ability for the public to buy stock. Any company that has shares traded on an open market like the New York Stock Exchange is considered a public company.
For EA to go private, a single individual, private company, or a private equity firm would have to buy out all of the available publicly traded stock, and then buy any of the private assets or holdings of the company too. It would be an enormous amount for a company like EA.
Going public has real benefits, and going private does as well. So going public usually means the company is getting a huge amount of cash to invest, it's an enormous fleecing of money that allows them to expand and grow, or, in many cases, pay off initial debts caused by early growth. The drawback to this is that the entire company is beholden to stock holders and investors, so if you or me buy $100 of EA stock, we have a say in the company and they have to report back to us ($100 is very little, we would have effectively no say, but let's say that you and I owned an investment firm and had $2m in EA Stock... we would have a big say in the future of the company). Legally, companies are beholden to their stock holders. It's the law, it's stock holder protection. If you invest in a company that is publicly traded, they are legally beholden to you, and it is illegal for them to do something to intentionally rip you off. That doesn't mean they can't fail or make bad products or go bankrupt, but it does mean that the CEO of the company couldn't, for instance, lie to you to get you to sell what you own only to turn around and do something that would make your just-sold stock increase. The benefit of being public is an enormous amount of cash: think of it like a Kickstarter in the millions.
The benefit to going private is that you are no longer beholden to investors. If you or I have a lot of say in EA, we might want them to make safe bets... Don't spend a lot of money developing new IPs because new IPs fail in sales 9/10 times; Don't spend a lot of time advertising for a game like Mirrors Edge because, c'mon, who's really going to buy Mirrors Edge after seeing a stupid commercial. Instead, we tell them, release Madden rehashes every year, copy CoD4 and release it as Battlefield 3... Make safe bets and get us good return on our investment. You and I are still good guys, we've bought a lot of stock, we're investing in videogames, but we also want a return on our enormous investment in the company, and we don't want them to do anything stupid like making overly aggressive games (think 38 Studios and their MMO project that helped sink them) and having us potentially lose our entire investment.
So, when the company is truly private, there are no outside investors, no share holders (you and me) that really don't know anything but want to make decisions, that control choices. Further, far less government regulation controlling who we can hire, how we can hire, where we can sell, how many taxes we have to pay, how we report our income, and so on. The company can do what they want and run the company how they want (generally). Finally, and most important, the company can store cash without reprisal. Apple is renowned for having a lot of cash, but they came under a lot of pressure by the SEC -- the body that oversees public companies and financial dealings -- for not sharing their cash holdings with their investors. This was a problem with apple for about 10 years. So finally, Apple started paying a dividend on their stock, basically, sharing a small percentage of their cash holdings with everybody who owns shares of the company. If you had apple shares, you'd get a deposit into your bank account simply for having the shares, without needing to sell your shares. The drawback to sharing a dividend, of course, is that maybe Apple was planning to use that cash to make the next iPad, and now they have less leverage to do so. As a publicly traded company, they've given up their right to manage their money truly how they want to, and both investors and federal regulators can tell them that they have to do something with it (in many cases, there are strong financial benefits to paying a dividend as well for the company... it has a residual benefit of increasing the value of a stock because people are more likely to hold onto their shares if they are getting a dividend, making the shares more valuable).
There are plusses and minuses to both. I work for a software company that has enormous sales, is very successful, and makes very popular products, but we're completely privately owned... not even owned by a private equity company or any initial investors. We're completely owned by ourselves. The company has never taken out a loan. It gives us remarkable leverage over our company and our choices and helps build on our success. At the same time, if we were to go public, it would make all of our employees rich immediately.
