A cookie grabber is any browser exploit on another website that allows an attacker to capture your browser cookies. Those cookies contain your login information, and can be used to log into your account from the attacker's computer.

The advice people in the forums give you is usually "delete your cookies!".
That will not make any difference if you've already visited the site with the cookie-grabber. They already have your cookies (which contain your password and deleting them from your system will make absolutely no difference to the scammer, it will only require you to log in again...
Once a cookie has been grabbed you need to change your password and e-mail, log out, then log in again. Deleting your cookies before you visit suspicious pages/outside Neopets sites, will help.

Another warning: if you downloaded a program from a neopets related site,or MSN/AIM and opened it, you could also get a virus on your computer which gives the hacker full control over your computer. He can do whatever he wants, like deleting files etc. Or it could be a Trojan Keylogger, which keeps track of everything you type. The scammer will know all your entry codes within minutes. So never ever download programs, and certainly not from people you don't know (even neofriends can become scammers, cause scammers never use their own accounts). Having a virusscanner on your computer is a pro. Also spyware is a way to get information about your computer, so you also should download Ad-Aware and Spybot to eliminate those

Watch out when you use chat clients such as AIM. A scammer is sending people links asking them to take a look at their 'artwork'. This image link is embedded with a trojan keylogger, so it will start recording everything you type, including your Neopets username and entry code.
Do not accept links or images or direct links from people you don't know via chat clients. This could be very dangerous, cause he doesn't stop at hacking your account, he will then hack into your computer and start deleting files, etc.
Tip: Set your AIM preferences for everything at "Display Approve Dialog" or "Reject."

More Info Here