I woke up one day thinking about how I'd describe Marxism to a friend who's only heard the term used as a bogeyman, and my wife encouraged me to write it down.
Marxism affirms a sort of Hegelian concept of progress: the idea that human civilization has, in general, gotten better over time and will continue to get better in the future. Marx argued that this progress has largely been driven forward by tensions that develop in response to economic conditions. Specifically, social and political progress have come from conflicts between the people who make their living by owning things and the people who make their living by working (generally for the people who own things).
For example, when you hear that a company exec only receives a salary of $1 but is somehow very rich, that's because they receive money for their part ownership of the company. It's great that they do work, but many owners don't work on the thing they own: if you had enough cash on hand, you could buy shares in a successful company and pay your bills just from the money you get by owning those shares, not because you work there. While some people are able to make a living by just owning things, the vast majority of people have to work for a living---and if they didn't, the world would grind to a halt. This dichotomy is inefficient at best and unjust at worst.
While things are better than they used to be, various conflicts of interest between owners and workers still come up. Marx predicted (in the 1800s) that civilization would eventually develop to a place where there would be no distinction between owners and workers at all (a period he called "socialism" or "communism," which were synonyms at the time). He didn't specify what that world would look like or how we would get there, and there's over 150 years of scholarship debating the topic. The thing they agree on is this: the interests of the owning class (bourgeoisie) and the working class (proletariat) are in conflict.
Remember that guy we met who retired early by converting two-story homes into duplexes and renting them out? He might do some paperwork and respond to complaints now and then, but he doesn't work full-time anymore; he's "retired." By owning stuff, he's able to opt out of working for a living, and most landlords are in a similar situation, where they just hire other people to do the work for them. I'll say here what I said then: if everyone did that, nothing would get done.
One of the main techniques in Marxist analysis is looking at how work and resources are really distributed instead of taking contractual language at face value. For example, some might say that people who take out mortgages don't really "own" their home, despite what the lender might say---if a borrower is 90 days late on payments in the 29th year of a 30-year mortgage, the mortgage lender will likely foreclose, and the "owners" will be out on the street with nothing to show for 28 years of payments and maintenance.
Along those lines, many freelancers register an LLC for the work they do, and any contracts they take classify them as an employee of their one-person business. Therefore, they proudly call themselves "self-employed." However, they are still selling their labor to businesses and thus have more in common with the working class than the owning class.