While the president has the power to appoint judges, the Senate reserves the power to approve or oppose the nominees. Before a future judge even has the chance to be voted on by the Senate, however, certain representatives get the chance to veto them.
Blue slips are papers that give home-state senators the chance to approve or oppose a nominee that would serve on the bench in their state. Without both senators supporting the nominee, they rarely make it to the Senate-wide vote.
Republicans removed this rule during President Donald Trump’s presidency, allowing for more judges to pass through the Senate. As President Joe Biden came into office, however, Democrats self-imposed the rule back, bringing some debate along with it.