You can move (transfer) the money tax-free to another IRA custodian, but the account must always remain an inherited IRA ("John Smith as beneficiary for Mary Smith") and must follow the 10 year rule. It can never be rolled over or transferred tax free into your own personal (non-inherited) IRA.
If you withdraw funds, you must pay the tax, and then it just becomes cash in your bank account and you can do anything you want with it, including making a new contribution to your personal IRA. But that new contribution must follow all the usual rules (you must have compensation from working, max of $7000/year or $8000 if over age 50, may or may not be deductible depending on your income, filing status and job situation.)
