Social Democracy is capitalism with a strong social safety net. They seek to uphold the triumvirate of private-public-state and make it as humanitarian as possible.
Democratic Socialism is socialism driven by voting/democracy. It is a subset or a particular implementation of socialism. In democratic socialism, there's no private sector. Only the public, public-co-ops and the state (which is itself beholden to the public, ideally). The key point is, no bosses but yourself and people of comparable power and authority as you with whom you voluntarily cooperate.
Most American self-styled DemSocs are SocDems in terms of policy. Elizabeth Warren is also a SocDem. AOC calls herself a DemSoc but I'm skeptical of it. She might be hiding her socialism power levels, however. There's evidence of that.
European nations Fox News describes as "socialist" are usually Social Democratic.
"But wait, what is socialism?"
Socialism is a "mode of production", which asks "how and why a society makes the things it does". It means when you're your own boss and companies are owned by the workers and they decide to make things they need or feel other people need. Imagine Amazon where there's no Bezos or board of directors and company policy is driven by workers. Like unions being the highest authority that exists in a company. Other modes of production: capitalism (boss and worker), feudalism (where you're born into the jurisdiction of a lord and you're beholden to them for life or until they're overthrown), slavery (people as commodity and ownership of people), primitive communism (prehistorical tribal society where everyone works for the sake of the tribe and everything is shared)
Autocratic socialism (Mao China, Stalin Russia, etc): Kind of an oxymoron but it's the way most of the western world views "socialism" and "communism". Autocratic socialism is when state power is concentrated in the hands of a few people and they decide everything for everyone else. As you can imagine, you can't really co-own a company if it's the state telling you that you own it. For people who call themselves socialists, autocratic socialism is when socialist revolutionaries degenerate into oligarchy. Except for tankies (Stalin apologists) but no one takes tankies seriously. For capitalists and socialism critics, autocratic socialism is the only historically extant form of socialism (arguably true, except for primitive communism) and why socialism can never work IRL. If you have a negative opinion of socialism, it's probably because you internalized this idea.
Other branches of socialism, not meant to be exhaustive:
Vanguardism: These are the people in the business of overthrowing stuff, usually violently.
Libertarian Socialism (Communism): These are socialists who oppose state power in all its forms and envision a world without nation states. Before "libertarian" shifted to describe right wing ultra-capitalists, it was referring to these guys.
Anarcho-Syndicalism: These are revolutionaries who seek to overthrow wage slavery with organized labor and unionism. Protests, strikes, all that good pro-labor stuff. Chomsky is one IIRC.
Municipalism: Socialists for whom the highest level of a state that should exist is the municipal government, aka locally democratic government, with inter-geographic coordination handled by municipal leaders.
Fully Automated Luxury Gay Space Communism: This is the society of Star Trek and what some (technological optimists usually) consider to be the natural end result of automation of labor.
Notable offshoots that are related but not socialism in the strictest sense:
Accelerationism: These are dudes who want to make everything that's bad about society even worse so that modern capitalism can meet its inevitable collapse that much sooner, so we can move onto the next phase of society, sometimes socialist, sometimes not. Imagine a creaky house. Accelerationists want to burn it down so they can build a new one.
Christian Socialism: This is socialism as interpreted through the lens of Jesus and the Bible. It is a theocratic approach to socialism
You shall not oppress your neighbour...but you shall love your neighbour as yourself: I am the Lord
He [the
Lord your God] executes justice for the fatherless and the widow, and loves the sojourner, giving him food and clothing. Love the sojourner therefore; for you were sojourners in the land of Egypt
When you reap in your harvest in the field, and have forgotten a sheaf in the field, you shall not go back to get it...When you beat your olive trees, you shall not go over the boughs again...When you gather the grapes of your vineyard, you shall not glean it afterward; it shall be for the sojourner, the fatherless and the widow. You shall remember that you were a slave in the land of Egypt; therefore I command you to do this
Give justice to the weak and the fatherless; maintain the right of the afflicted and the destitute. Rescue the weak and the needy; deliver them from the hand of the wicked
— (Ps. 82 (81): 3, 4).
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Blessed is the man who fears the Lord, who greatly delights in his commandments!...He has distributed freely, he has given to the poor; his righteousness endures forever; his horn is exalted in honour
— (Ps. 112 (111): 1, 9).
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I don't consider it socialism in the Marxist sense because Marxist analysis tends to be materialist (he believed only in the physical world and its laws). That said, there's no reason some forms of Christianity (as well as Buddhism, or Shintoism or other types of egalitarian spiritualism) should be incompatible with socialism.
(Disclaimer: None of this is meant to be definitive. I'm simply providing my understanding of "socialism", which is a broad idea, for those who are curious but don't know what questions to ask, or those who naturally seek knowledge and understanding. The problem with "socialism" as a movement is that there's a hundred branches, each with their own specific ideology. This is also its greatest strength.)