After Googling and reading through all the usual reasons for this error when trying to connect to a MySQL database I was starting to get frustrated because I could connect to MySQL server using several GUI applications so knew my root password was correct. The problem only cropped up when I wanted to launch MySQL from a bash script to execute a stored procedure via a cron job. When I stepped back and took a look it turned out to be a fairly obvious problem, I was trying as per this example (obviously not the real password):
mysql -u root -pABC$123
I normally like to include a few special characters in my passwords and use more than eight characters to negate the possibility of them being cracked using reasonable length rainbow tables. Of course the dollar sign was causing bash environment variable substitution so it was simply a matter of prefixing the dollar sign with a backslash:
mysql -u root -pABC\$123