It is commonly the memory card that is failing, and the raw data on the card is corrupted, but it could be any other hardware component, as Keith suggests.
To narrow down the area of the problem, recopy the photos, perhaps by hand using Explorer or Finder, to the hard-drive of your computer, and reimport this second copy of the photos, having temporarily turned off duplicate suppression in the Import panel.
If the same photos are corrupted in the same ways then the camera or memory card is likely the problem and you should discontinue using that memory card, and try a new one. If the problem persists then the camera may be the problem.
If different photos are affected from one import to the next then something in the transfer or on the computer (Keith's situation with the HD) is likely bad. You can try transferring the photos a different way--using a card read instead of a cable on the camera, or getting a new card reader if you're already using one.
A problem with the hard drive as Keith said he had can be much more difficult to fix but also very important to fix since you're losing photos and perhaps other data as the drive spins.
Sometimes the memory on the computer is bad, but usually the symptom is thin lines or tiny spots of color in many images, not large portions of images missing.