Senate Dems were referring to President Bush's use of presidential "signing statements." These statements generally contain instructions on how the executive branch intends to implement legislation that Congress has passed and the president has signed into law.
But, as the Financial Times explains, these signing statements frequently contain language that reflects the White House's desire "to ignore congressional intent." As the FT article noted:
The issue has attracted scrutiny following a study by the Boston Globe this year that found that Mr Bush had employed signing statements for more than 750 laws enacted by the Republican Congress.
“That’s far more than all the signing statements signed by every single president from George Washington to Bill Clinton, put together,” said Patrick Leahy, a Democratic senator, during the hearing.
Concern has focused on whether Mr Bush has used the low-profile statements to side-step congressional authority, rather than publicly issue a presidential veto of legislation.
Mr Bush has never issued a formal veto.
“Basically, the president signs laws enacted by people’s representatives in Congress but he’s crossing his fingers behind his back. And when he says he never had to make a veto, heck, why? He just signs laws saying he is not going to follow them,” Mr Leahy added.
The most controversial use of a signing statement came this year after the Senate voted to pass a prohibition on certain kinds of interrogation procedures. After lengthy negotiations with Senator John McCain agreeing the terms of an amendment, the administration “issued a signing statement which appeared to undercut what had been negotiated”, said (Republican Senator Arlen) Specter.