My mother is on disability, and doesn't file taxes. She recently won dental services. She's going to receive a 1099 for about $45k from the dentist. How does this work?

It is unclear exactly what you mean by "won".  But this presumably means that she "won" something like a prize.  This would be considered income to her, and based on what you are saying, it would likely be reported as 'Other Income' and reported on Line 21 of Form 1040.  In that case, your mother would be required to file a tax return, and pay taxes on the amount received.

Yes she won a contest for much need dental work. Anyway I can figure out how much is will owe? She's on a fix income.

It is not clear, because if she is receiving disability income and/or other income those amounts must be considered when determining how much she owes. Also important is her age and filing status (single, married, etc.)

And check with disability as it may also effect the benefits she gets.