If someone leaves me a house in their Will but money is still owed, what happens next?

A mortgaged property can be inherited just like any non-mortgaged property. However, normally, when a property changes ownership the mortgage lender has the right to demand full payment of the entire loan balance. Most mortgages have what are known as "due on sale" clauses; these clauses allow lenders to call the mortgage balance due if the home changes hands. However under federal law, inherited properties are exempt from these clauses; lenders can't call a mortgage due if the home transfers either to a joint tenant, to a tenant by the entirety or to a family member under the terms of a Will. The mortgage stays in place and the person inherits who has to make the payments.
Further, some Wills address the issue of any outstanding mortgage. It may direct the executor of the estate to pay off the mortgage loan with other estate assets. In that case, the inherotor would receive the property free and clear. Otherwise, they must make the mortgage payments themself (as stated above). The lender might require the person who inherits to officially assume the loan, personally taking responsibility for it, but it can't deny them the right to keep making the payments. This means that it can't force the inherotor to refinance the mortgage into thierr own name. In fact, they can keep the property in the deceased's name if for some reason they wanted to. 
That having been said, mortgages on homes inherited by non-relatives or non-joint owners usually can't be assumed. In that case the inheritor generally needs to make arrangements with the lenders to pay off the mortgage (e.g. refinance, etc.).
The process for relatives taking title and recording an inherited property's deed are generally the same as for non-relatives. You can consult directly with a real estate attorney for advice pursuant to specific state law. Also, consulting with a lawyer is a good idea in such a case since mortgaged properties can come with property taxes owed, liens attached and even estate taxes due.