Use the IRS Interactive Tax Assistant to find out
https://www.irs.gov/help/ita/whom-may-i-claim-as-a-dependent
There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). The Other dependent (qualifying relative) credit is worth (up to) $500 per dependent and is non-refundable. That is, it can only be used to reduce an actual tax liability.
A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he meets the 6 tests for claiming a dependent:
He must have a US social security number or tax identification number (TIN)
Nontaxable Social security disability doesn't count as income, for the income test, but social security money he/she spends on her self does count as support not provided by you, for the support test. Money she puts into savings does not count as support she spent on herself. Note that a grandparent is closely related so there is no requirement that she live with you at any time, during the year. But if you provided a home it helps your support case, unless they own the home you live in. If no one person (or married couple) provides 50% of the support (for example your siblings are also sending support), then a "multiple support agreement” (IRS Form 2120) can be used, to allow you to claim the dependent. https://www.irs.gov/pub/irs-pdf/f2120.pdf
The IRS has a worksheet that can be used to help with the support calculation. See: http://apps.irs.gov/app/vita/content/globalmedia/teacher/worksheet_for_determining_support_4012.pdf The support value of a home is the fair market rental value, divided by the number of occupants.