Lets define some terms, this will be important later (definitions from Merriam-Webster):
Socialism: any of various economic and political theories advocating collective or governmental ownership and administration of the means of production and distribution of goods. (The people as a whole own the means of production)
Democracy: government by the people; especially : rule of the majority
One of the main points of confusion surrounding socialism is that while it is an economic system, it is often used incorrectly to describe the political system of a state. A country can be socialist and democratic in the same way that a country can be capitalist and democratic. At least in the United States, the terminology was complicated by the political struggles during the Cold War in which socialism (and communism) popularly came to be identified with the authoritarian political systems of the USSR and its allies.
A democratic socialist therefore is someone who advocates for the economic system of socialism through the democratic process. (This contrasts them from more radical socialists who believe that socialism can only be established by revolution (contrasted further with Karl Marx, who did not see socialism as an end goal, but rather a state of transition required for a society to reach communism)).
A final point of contention is the difference between Democratic Socialists and Social Democrats. Generally speaking Social Democrats want to democratically reform capitalism with greater government intervention, rather than fully replace it.
The United States has never had a president who advocated for the end of Capitalism or one who self-identified as a Democratic Socialist.
The most famous Socialist in American history is undoubtedly Eugene V. Debs who led the Socialist party during the early 20th century. He was the Socialist candidate for president 5 times, receiving ~6% of the vote in 1912 (his best performance).