To be an accountable nonprofit is to be responsible and answerable, not only to the people they serve but also to the general public (taxpayers).
A non-profit organization does not pay certain taxes to the federal government and in some cases to state and local governments. Specifically, it doesn't pay corporate income taxes to the IRS if it operates solely for the public benefit. The organization must file an annual 990 reporting form to the IRS; it must also state any changes in its purpose & activities and must submit to periodic audits to ensure that it continues to adhere to standards. In exchange, the charity will not be taxed on its income, allowing it to use those funds toward its cause to meet its stated goals.
Additionally, donors who itemize can take a tax deduction for their contribution to charities.
In 2014, total giving by Americans to charities was $358.38 billion which was tax free income and donors may record their donations as a tax deduction on their itemized income tax return.
While charities receive significant tax advantages, there’s very little oversight over how they spend donations. A large number of nonprofits have ineffective strategic plans. The services they are supposed to provide may be inappropriately funded.
According to the Association of Fundraising Professionals, an accountable nonprofit is responsible for:
Leadership on behalf of the public interest
Enhancing the well-being of communities and society.
Promoting inclusiveness, pluralism and diversity within society.
Educating the public, business, nonprofit organizations and government, including appropriate advocacy.
Maintaining effective governance and management.
Generating adequate resources, managing resources effectively, supporting and recognizing volunteers, and appropriately compensating staff.
Avoiding conflict of interest and abuse of power.
Here is a good article on nonprofit accountability: Nonprofit Accountability and Ethics: Rotting from the Head Down