Q:

When I try to enter a user password to delete an account, it doesn't work. and I know the password I'm using is correct

I got a new computer today. I my parents logged in with a username I didn't want so I made another admin. Now, I'm trying to delete the old account. 
when I go to Users & Groups in System preferences, click on the old account, hit the - sign to delete it and enter the password, the popup shakes like the password didn't work. I'm sure the password is correct because I used it multiple times to log into the account. I tried switching the old account to admin but that didn't work.

A:

If you can't delete the user for some reason, you can remove it at the command line.
Launch "Terminal" (from /Applications/Utilities/Terminal.app, or search via Spotlight) and type:
sudo dscl . list /Users

...followed by your administrator password.  Then hit enter
The output will show you the shortname of your local users (towards the bottom).
Here I called the user to be removed "testy" (I'm only showing the last few entries):
...
admin
daemon
Guest
nobody
root
testy

Now remove the user from your system and from System Preferences (exchanging the shortname "testy" with your own, found earlier):
sudo dscl . delete /Users/testy

Then hit enter.
The home folder of that user you can either keep, if there are important files, or delete it with...
sudo rm -r /Users/testy 

...and again hit enter.
PS. A speculative answer for the permissions error received and mentioned in the comments section can be found at StackExchange:
What causes eDSPermissionError when running dscl?

