Democrats said their long-pledged push to tighten the country’s gun laws would take center stage early this year as lawmakers introduced legislation Tuesday aimed at expanding background checks to nearly all gun sales.

House Speaker Nancy Pelosi (D., Calif.), Rep. Mike Thompson (D., Calif.), who leads a task force focused on reducing gun violence, and former Rep. Gabby Giffords (D., Ariz.) officially unveiled the legislation expanding background checks to more commercial sales and private transfers, with the goal of flagging...