Is there a word for numbers between 10 and 99?

Assuming you mean 10-99 inclusive, the phrase you want is "two-digit number".

Numbers from 10 to 99 inclusive are often referred to as double figures in the UK.

For example, from yesterday's (British) news:

The death toll during the recent cold snap has hit double figures - as forecasters predicted up to another foot of snow is on the way.

There isn't any context in the question, but if the context is something like

The number of people at the meeting was between 10 and 100

then the phrase to use is in double figures:

The number of people at the meeting was in double figures.

Commonly used would be "two digit" or "double digit".

• "Please enter a two digit number"

If you REALLY want to exclude 10 you could say

"Please enter a 2 digit number greater than 10"