Your browser is deleting cookies and/or stored passwords whenever you close it (or, alternatively, you've got some security software which is doing the same thing). You need to go into your browsers settings to change its behaviour but, without knowing which browser you're using, its hard to advise you more specifically.
There have been so many fiddles on ebay, (with passwords locked in) they have now made everyone enter the password for every transaction.
It will only remember your user name.
Talking of Ebay, have they stopped the alerts when the end of an auction is near? I thought it may be because I now use Firefox instead of IE, but does anyone still get them? They were so useful for forgetful people like me!