Assuming your mother's name is the only name on the title, then:
Your mother is selling her home (legally speaking). If she owned and lived in the home more than 2 years, she can exclude the first $250,000 of capital gains, and any other capital gains are taxable. Capital gains is the difference between the selling price and the adjusted cost basis. The adjusted cost basis is the original purchase price of the home, plus allowable adjustments, plus the costs of permanent improvement (not repairs), minus depreciation that was claimed or could have been claimed if the home was used in business (home office, home daycare, rental). If your mother bought the home with a spouse who has since died, that can be a significant adjustment to the basis that will lower her capital gains. See publication 523 for more details, and post more details here if you want a more specific comment.
https://www.irs.gov/forms-pubs/about-publication-523
Then, legally speaking, your mother is giving a gift of money to her daughter. If the gift is more than $17,000, it must be reported on a form 709 gift tax return that the mother must file. However, gift tax is not actually owed unless the mother's total lifetime gifts plus estate are more than $12 million. But the form 709 is filed so the IRS can keep track of large gifts.
However,
There are other implications that need to be discussed. You and your sister need to see a legal firm that specialized in estate planning and elder law. For example, if your mother needs to go on Medicaid to pay for long term care, her Medicaid eligibility is reduced by any large gifts she has made within the past 5 years. Your sister could be forced to pay the gift back, or else your mother won't get Medicaid coverage. (Medicare pays for doctor and hospital visits, but not nursing homes or assisted living. You need Medicaid for nursing home care, and to get Medicaid you must be "poor", and that means that if you give away your money to count as "poor", the government can take those gifts back to pay for care.)
There are other implications to your situation. Proper estate planning can preserve assets and make sure your mother has the best possible quality of life.
