DRE's are very common problem with the first couple of generations ps2's, sometimes cleaning wont help and you'll have to replace the laser ( $30 hardware ). I've done one before, I've also cleaned one before without replacing the laser as well. You should be able to clean it and it work, if not, the laser is burnt out and will have to be replaced. I can do the replacement for $20 + $30 for hardware, you might want to ante up and just by a new one by then.
If you can look at the bootup screen ( when you dont put a cd in, the screen that pops up ), you can get a hardware model number, it should be like SHC or scp ( acronyms are killing me right now ), tell me what it says and I can find out how old your ps2 is and if it is affected by the common DRE probs.