When Bill Clinton was running for the presidency in 1991-2, the Democratic Party was in the midst of a major decline. Only one Democrat had been elected president since Lyndon Johnson won election in 1964. Jimmy Carter won in 1976, partly because of the Watergate scandal, but lost his reelection bid in 1980. That meant that Republicans had won 5 of the last 6 elections. It was in this context that Clinton proclaimed himself to be a "new Democrat."
What he meant by this is that he was not a typical liberal Democrat. Instead, he positioned himself as a centrist. He symbolically showed his freedom from the Democrats' black base by attacking a rapper, Sister Souljah, for saying that blacks should have a week where they killed whites instead of one another. As president, he pledged to end "big government as we know it." He led the fight to reform welfare even though welfare had been something that liberal Democrats had fought hard to protect when Pres. Reagan attacked it.
In ways like these, Clinton tried to show that he, and the Democratic Party, had become more centrist and moderate and that they were, therefore, worthy of being elected.