Since I established my Yahoo mail account two years ago, my login and
password (which have never changed) are rejected when I enter them on
the Yahoo mail sign in page. A new screen pops up saying "invalid
password." When I re-enter the same information, it is accepted and I
can access my mail. I have to go through this every time I check my
mail, ten times a day. Why do I get an "invalid password" message on
the sign in page, but not on that next page asking me to re-enter?

Have you tried deleting your internet cookies?

Your cookie expired!

As david1977 suggested, you need to dump your Yahoo! Mail cookie and
get a new one.  All you need to do is go to your cookie folder and
sort them by name.  Look for anything having to do with Yahoo! and
delete them.  Or you can delete them all if you don't want to look for
it.  Then-- the next time you sign in, Yahoo! Mail will deposit a new
cookie on your computer, good for another couple years.

If you delete ALL your cookies, understand you will be having to
re-enter user names and passwords at all the sites you were holding
cookies for.

~~Cynthia

Hi there..

Greetings from india.
Sometimes proper settings in internete explored may cause cacing problems.
 
In you internet explorer.

go to 

tools-->internet options-->click on a tab called settings-->
ensure that "every visit to page" the first option is selected and
give you whole thing a try again.

Please let me know how this goes.

Ashish