By "upgrade" he means a new laptop with a more powerful graphics card like the nVidia GT 660m which is really the minimum laptop graphics card to effectively play games at 1920x1080. Not sure what AMD Radeon Mobility HD graphics card is equivalent to it.
99%+ of laptops does not allow you to upgrade the graphics card. That small fraction of 1% that do are specific gaming laptops that generally cost over $1,500. Additionally MXM card (graphic card format for laptops) are not very easy to find and are expensive.
All games are designed differently, but the simple fact is that the higher the resolution you play at the greater the demand on the graphics card because it must "draw more things on the screen". If you think Assassins Creed 3 is demanding, wait until you try Metro 2033; you'll get a slide show.
Also everyone has different expectation. Some people only want to play games at high or ultra graphic quality. If you can settle for low graphic quality then you don't need something quite as powerful as the GTX 660m I suggested. However, the GT 630m is a bit too weak to play many games at 1920x1080 resolution.
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