Asus expects to sell a lot of tiny computers this quarter. The company shipped 700,000 Eee PCs during the first quarter of 2008, and is predicting sales of 1.2 million or more Eee PCs during the second quarter.

That prediction makes sense, since the company’s first ultraportable has gained a lot of attention over the last few months. When it was first launched, it was a computer that only ubergeeks had really heard about. While I’m not sure that the Eee PC has become mainstreamed just yet, I run into an increasing number of people at coffee shops who actually know what that little thing I’m typing on is. Add the fact that the company expects to ship its updated Eee PC 900 in Q2 with a larger screen, more memory, and the option of purchasing a Windows XP model, and Asus should easily be able to top its Q1 sales.

But come Q3, it’s anybody’s guess what will happen. While Asus is expected to update its Eee PC line yet again this summer or fall when the next generation Intel Atom processor is released, Asus won’t have the market all to its lonesome anymore. The computer is already facing competition from the HP Mini-Note and a variety of computers built on the VIA Nanobook reference design, including the Everex Cloudbook.

But soon Dell, Acer, Gigabyte, MSI, and other computer makers will be bringing ultraportables in the $500 price range to market as well. And while Asus is certainly making a name for itself, many of these companies have already made a name as providers of mainstream laptops. When Dell or Acer launch a low-cost ultraportable, it won’t just be the ubergeeks who find out about it. It will be anybody who’s ever bought a computer from that company and receives promotional mail or email from the company. And while you can already walk into specialty stores like J&R in New York City and see the Asus Eee PC on display, when customers can walk into a Best Buy or Circuit City and see a whole array of tiny, cheap laptops, I think a lot are going to gravitate toward the brands they’ve heard of.

Of course, if Asus continues to release high quality, low cost computers that continue to get excellent reviews, Asus could be one of those brands that people have heard of by the time the fall rolls around and people are headed to their electronics superstores to pick up those back to school laptops.