In the eyes and barbed language of his enemies, Zohran Mamdani, the front-runner for mayor of New York City, is the Democratic nominee in name only.

President Trump has called him a “100 percent communist lunatic.” Former Gov. Andrew M. Cuomo, who is running as a third-party candidate and polling second, has sought to highlight Mr. Mamdani’s roots as a democratic socialist, suggesting that his positions are “dangerous” for New Yorkers.

Mr. Cuomo used both of his appearances on Fox News last week to criticize Mr. Mamdani’s left-leaning views, warning that they would be a “death knell for New York City.”

“New York City people are not socialists,” Mr. Cuomo said.

Neither, actually, is Mr. Mamdani: He is a democratic socialist, which means his beliefs are similar to those of socialists but not exactly the same. He is a member of both the national Democratic Socialists of America and its local New York City chapter.

On Thursday, however, Mr. Mamdani said that his mayoral platform was not the same as that of either the local or national D.S.A.

Need more clarity? Here’s a breakdown of how socialism and democratic socialism work, and where Mr. Mamdani fits in.