What happens to the equity I have in my home when I refinance?

The payoff amount on my current mortgage is $ 88,000 on a house I bought for $ 140,000. If I refinance will it be for $ 88,000?

It doesn’t seem right since I will appear to be halving the dollar amount of my monthly payments by just reducing my interest rate by 1%.
Details I omitted from original question for clarity: I have 22 years left on a 30-year fixed rate at 6.125%. I am refinancing due to a divorce and I desire to retain ownership of the house. I do plan on staying in this home for a long time, and, due to alimony I must pay, want the lowest monthly payment I can get.

Thanks for the great answers guys.

I

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