I have the HP Deskjet D2660 printer. The new cartrages are the correct ones. I peeled the tape off and inserted them correctly, but the no ink light keeps flashing and it won't print anything. The printer is on line, and when the printer sits idle for ten minutes or so the light goes out, but the minute I go to print it comes back on and won't print anything.

I have to ask this question of you, just to make sure this is not an issue. Are you using a factory new ink cartridge?
Please consider the following:
If you are using a refilled or third party cartridge that may well be your problem. Most manufacturers build logic into their products to prevent the use of “Other Equipment Manufactured” cartridges as well as re-filled cartridges.
They do this to prevent problems from contaminated inks causing problems with print quality or damaging the physical printhead.
If you are using factory new cartridges, then please call the manufacturers Technical Support for assistance with the warranty on the cartridges