Attorney General Jeff Sessions met with Russian diplomats including Russia's ambassador to the U.S. in July and September 2016, and then told the Senate Judiciary Committee under oath in his confirmation hearings that he did not have communications with Russians at that time. If Mr. Sessions intended to lie, this would be a clear violation of the law, including potential charges of perjury and obstruction of justice.
Explanations for not disclosing this information include not recalling the meeting and interpreting the question as pertaining to only political activities. When asked about the event, Mr. Sessions said, “My answer was honest and correct as I understood it at the time.”
Since this event has come to light, Mr. Sessions has recused himself from the Justice Department's ongoing investigation into Russian involvement in President Donald Trump's campaign. The fact is that this breach should result in Sessions' resignation, or if not, impeachment.
While perjury can be difficult to prove due to the requirement of proving intent (See Penal Code Section 118), if Sessions is found to have committed perjury, he would properly be charged with a felony and likely be impeached by the House of Representatives. The repercussions of such an investigation and potential conviction would affect more than just Mr. Sessions, but could affect Donald. Trump's presidency as a whole.