Checking a cow’s teeth is a way to get an idea of how old she is if age is not known. The classifications that you asked about are commonly used when selling beef females, especially at a sale barn, where the age may not be known.
A “solid” mouth cow would mean that the cow has all her teeth, which would mean that she is a young to middle-aged cow. A “broken” mouth cow would indicate that the cow is starting to lose some teeth, an indication that she is an older cow. A cow that is called “smooth” mouth would mean that she has lost all her teeth and is therefore considered old.