The most likely problem is that your power supply isn't good enough. Chances are you just have the standard 240W power supply, which isn't enough to reliably power a 150W graphics card like the 660ti. You'll need to upgrade to the 330W power supply available from Dell/Alienware.
The other likely problem is that it might just physically not fit. To check, you'll need to open up your case, take out the graphics card, and measure the space available for a replacement.
This http://www.newegg.com/Product/Product.aspx?Item=N82E168... is the smallest 660ti; it's 7.53" x 4.38" x 1.39" and requires two free expansion slots. If the card in your X51 is a single-slot, low-profile, or short card, you're going to be very limited on upgrade options, and the 660ti won't be one of them. From looking at some pictures and reading some reviews, it appears that X51s use dual-slot full-height cards, but it seems like the max length is 9 inches; that rules out every 660ti I can find except for the Zotac.
If you have the power and you have the space, it's unlikely that you'll have any other issues. It *appears* from what I'm reading that the X51 uses relatively standard components (aside from the power supply) and is designed to be upgradable (to a point). So it should take any PCI-E card that fits and doesn't require too much power.