The basic facts: This story is about the sale of a controlling stake in a Canadian company called Uranium One to Rosatom, the Russian atomic energy agency. Because Uranium One controlled uranium mines in the United States, the sale had to be approved by the Committee on Foreign Investment In the United States (CFIUS), part of the executive branch.
A number of investors in Uranium One gave donations to the Clinton Foundation during the time the sale was being considered (between 2008 and 2010), in part through the participation of Frank Giustra, a Canadian mining magnate who was a large donor to the Foundation and who had controlled a company that eventually bought Uranium One (according to the Times, Giustra sold his interest in the company in 2007, before the Rosatom deal).
In addition, Bill Clinton was paid $500,000 in 2010 to give a speech to a Russian bank with ties to the Russian government. The U.S. government eventually approved the deal in 2010.
What's the allegation against Hillary Clinton? The reason this is a story is the potential that there was some quid pro quo involved: that in exchange for donations to the Clinton Foundation and/or the speech Bill Clinton gave in Russia, Hillary Clinton used her position as Secretary of State to make approval of this sale happen. It need not be explicit, but at the very least there has to be a connection between donations and official action that Clinton took.
What's the evidence for that allegation? There isn't any, at least not yet. The only evidence is timing: people who would benefit from the sale made donations to the foundation at around the same time the matter was before the government.
What's the evidence in Clinton's favor? Even if Clinton had wanted to make sure the sale was approved, it wouldn't have been possible for her to do it on her own. CFIUS is made up of not only the Secretary of State, but also the secretaries of Treasury, Justice, Homeland Security, Commerce, Defense, and Energy, as well as the heads of the Office of the U.S. Trade Representative and the Office of Science and Technology Policy.
The Director of National Intelligence and the Secretary of Labor are non-voting members, and CFIUS's work is also observed by representatives of other agencies like the National Security Council and the Office of Management and Budget.
The idea that Clinton could have convinced all those officials and all those departments to change their position on the sale, even if she had wanted to, borders on the absurd.
Furthermore,
the official who was the State Department's representative on CFIUS at the time, Jose Hernandez, told Time magazine that Clinton did not participate in the evaluation of this deal: "Secretary Clinton never intervened with me on any CFIUS matter," he said.