Americans here: what is right wing, what is left wing?

The fundamental differences, once we put aside issues of identity (which are fraught with tribalism), are all about value preferences.

Left-wing value preferences (generally):

• Equality in fact is more important than equality in law, especially to address historic inequality in law.
• Personal autonomy is more important than a lot of things, especially a sense of sanctity of human life, national identity, national autonomy, and personal safety.
• Personal autonomy is less important than securing everyone's basic needs, issues of justice (defined using these other value preferences), and sustainability.
• Positive freedom (the ability to do something) is more important than negative freedom (the freedom from some other thing) alone.
• Looking for the best new way forward is more important than retaining what was best about the old way.
• The public good is more important than private good, which leads to awkward feelings toward corporations.

Right-wing value preferences (again, generally):

• Equality is law is more important than equality in fact.
• Negative freedom (the freedom from) is more important than positive freedom (the freedom to).
• Retaining what was best about the old way is more important than insisting on changing to the best new way.
• Personal autonomy is more important than a lot of things, especially guaranteeing everyone's basic needs (which seems to be in opposition to "personal responsibility" for many right-wing thinkers) and sustainability.
• Personal autonomy is less important than the sanctity of human life, national/ethnic identity, national autonomy, issues of justice (defined differently from the left-wing sometimes), and personal safety.
• Personal safety is especially important to the right-wing thinker, as is personal responsibility, faith, stability, and duty to one's people.
• The private good is a public good, which leads to lots of awkward feelings about corporations and distrust of government intervention into them.

There is a lot more overlap besides the points I've brought up that distinguish the left wing from the right wing. For instance, all left and right wing thinkers in the US are anti-monarchists. They all highly value individualism in the sense of personal agency. They all care deeply about helping the worst off, though they vary significantly in how important the problem is, who the most vulnerable are, and through what mechanism to help the most vulnerable. They are each wholly comfortable using government authority to act on the values that are most important to them, even when those values are in opposition to the other wing's values (sometimes, especially so). What I focused on in this answer was the areas where they value distinct things, not their shared traits.

It's also important to note that these preferences aren't saying that left-wing or right-wing thinkers are against the things that they value less than others. Values, by their nature, are generally seen as good things by nearly everybody. The fight often boils down to what values are the most important, not whether a value is a good or bad thing.

I was fortunate enough to have an educator in college express some concepts to me about the conservative (the "right wing" party in the United States) part of the US that helped make sense to me as someone who has far more liberal ("left wing") views. One of the keys to the expression that was presented to me, was that you can imagine the political spectrum as a circle, because as soon as you reach the extreme of either spectrum, you start see more similarities between them than you do differences. The content changes, but the approaches start to sound more and more similar. (And this is apparently being expressed as the horseshoe theory of political theory, which another poster correctly identified above!)

It's far easier to define what a conservative is "against" as opposed to what they "for." It's important not to laugh at that concept, because conservative could be best described as resistance to change. The conservative wing is against the prospect of gay marriage because it's changing the common concept of marriage being between "one man and one woman." You can see a similar resistance to concepts like polygamy, polyamory, or any other non-tradition form of relationship. This isn't because conservatives hate gay people, or because you can't possibly love two people. This is rooted in the idea of "Why do you need to change this? Why aren't things acceptable as they are?" The maintenance of the current or long-standing norms, this is the core of the conservative politic viewpoint.

However, in the US, we suffer the consequences of realistically only having two political parties, and a democracy effectively designed to only utilize two political parties. This has forced any non-Republican or non-Democrat party to either collapse into the party that most resembles their point of view, or to effectively be doomed to irrelevance in the political landscape. The Tea-Party phenomena of the early 2010's really exhibits this. They collapsed into the "far right" wing of the Republican party, even if their views are realistically more in line with a stereotypical Libertarian political view. This, as a result, hybridizes the Republican party, forming a sort of coalition, wherein we see aspects of the different segments of the party collide, or even actively fight against themselves. (Again, see the lame-duck sessions of the US Congress during the 2010's.)

That said, the current Republican party consistent of both fiscal and social conservatives, religious evangelicals, defense hawks, and neoconservatives. They have a strong focus on lack of large government structures outside of those defined in the US Constitution, and generally believe that the best federal level government, is a government that is less involved in their day-to-day lives.

The current Democratic party consists more traditionally Liberal (note that I'm not referring the lowercase "liberal" as in "more free" term. Liberalism in the United States is pretty well detailed here: ( https://en.wikipedia.org/wiki/Modern_liberalism_in_the_United_States ) and I'm referring to the Liberalism defined here: https://en.wikipedia.org/wiki/Liberalism )) leaning aspects of the American population. This means a coalition of social, religious, identity, and fiscal egalitarians. The modern Democratic party seems to be more focused on the equality of as many different groups of people as possible. The Democratic party generally believes in the active role of the federal government to participate in the lives of the citizens, in order to generate the equal opportunity of as many people as possible.

I didn't specifically try to state many stances, because I think it's important to understand that each member of a political party, particularly in the United States, will have wildly different views due to the inherit need of any person to really partake in a political party in order to actually participate in the political process. Independents, while they exist, are rarely capable of generating enough capital to be elected in the modern environment.

The more I think about it, the more I come to the conclusion that "Left" and "Right" don't actually mean anything. At least in terms of US politics. I mean, we can't even properly define the terms, how can they possibly be useful in discourse?

With only two political parties in the US, they and their platforms are the effective definitions that you are looking for. But the problem is that political platforms are complete bull; they never accurately describe any particular politician's viewpoint, being written instead with the express purpose of convincing people to vote for them.

Very few people vote for a candidate because they like what they say or do. I would go so far as to say the majority of voters has never heard from nor read of a great number of the candidates they vote in for office. They vote for the candidate because they want their tribe to win. And this tribalism is a huge problem. This tribalism allows a number of political distortions that are generally much less powerful in other nations. Our "conservative" party is more obsessed with dismantling the government and civil protections than it is in protecting our economy or culture, and our "liberal" party is too conservative to pass any meaningful legislature.

Of course, the most damning part of this American tribalism is that the logical conclusion of the increased polarization caused by it is authoritarianism. And as proof I present the American president, who at the very least seems to project the airs of a leader of a fascist state. And this isn't only a Republican problem; Nancy Pelosi is still loved by many Democrats in spite of her many flaws, and I would say that one of the deciding factors that encouraged a large swath of voters to choose Hillary Clinton over other candidates was because she was next in line in a (largely imaginary) Clinton dynasty.