How do seniors citizens specifically a person 80 years old determine if they have to pay taxes?-- My father SSI is 1000 month and he gets a pension that is equal.

Unless you have left out some information about your dad's income, it sounds like no, he would not need to  pay federal income tax.  If he is having tax withheld, he might get a tax refund by filing a tax return.  (And he should receive the stimulus check from the federal government.   With the pension he receives, he cannot be claimed as anyone else's dependent.)

Age actually has nothing to do with it.  Income determines whether a person needs to file a tax return.  You say your father receives "SSI"  but is he actually receiving Social Security benefits?  SSI and Social Security benefits are not the same thing.  Which does he receive?  If he receives a SSA1099 then he gets Social Security.  If he gets SSI, he does not receive a tax form at all for it.

WHAT IS THE DIFFERENCE BETWEEN SOCIAL SECURITY BENEFITS AND SSI?

https://www.ssa.gov/ssi/text-over-ussi.htm

And....paying on a mortgage does not change his tax situation unless he has tax withheld, and has enough itemized deductions to exceed his standard deduction.  

You must file a federal return if half your Social Security or RRTA benefits plus your other income exceeds

Single, HOH, or qualifying widower  $25,000

Married filing Jointly  $32,000

Married filing Separately $0

Who has to file?

http://www.irs.gov/uac/Do-I-Need-to-File-a-Tax-Return%3F