It sounds like something's going on with your RAM. When the computer has a RAM issue (like the RAM's failed, or the computer can't detect it, or it can detect it but it's an incompatible type), hitting the power button causes one or more beeps, sometimes accompanied by a blinking sleeplight:
Power On Self-Test Beep Definition - Part 2
Intel-based Mac Power On Self Test RAM error codes
One beep = no RAM installed. Have you been playing arond with RAM recently, or has the computer gotten knocked around?
The first thing to try is to reseat the RAM correctly:
Installing MacBook Core 2 Duo RAM
The process for installing/removing RAM on the earlier A1181 Core Duo MacBook is identical. I like to use a pink pencil eraser to rub the RAM contacts clean of scuzz, and use a soft, dry toothbrush to brush the rubber crumbs off the RAM contacts and clean the inside of the RAM socket. Then reinstall, reassemble and reboot, and see if anything changes.
If that doesn't work, the next step is to replace the RAM - preferably with sticks that you've seen working in another computer, so you can be sure they're good.