The following is my best-effort explanation of Buddhism. I am no expert, and I mostly wrote this as a record of my understanding as it is right now. If I have something wrong here, please let me know.
Buddhism, to some extent, makes predictions, challenges you to falsify it, and accepts your counter-arguments. In the terminology of Buddhism, it's a "gem-in-the-hand" belief: there's nothing held in the closed fist of the teacher, there's no "gotcha" or thing you must take on faith. Buddhism doesn't really have "faith", you can sort of agree with it or not agree with it.
Some parts of Buddhism can simply be told to you, in the same way that I can tell you that the Boston Tea Party happened in 1773, and now you "know" that fact. Other parts require practice to understand, in the same way that explaining conceptually how to juggle does not literally make you able to juggle.
About Meditation - In your head, there's a little voice. You can make it talk if you want. Here, make it say "purple elephant!" That's the voice I'm talking about.
You're the thing hearing the voice.
Meditation helps you see that little voice and see you're the one listening. Many people seem to think meditation is a calming exercise. It's not. It can be calming, but the "point" is more to notice that you're not actually the one thinking.
After all, if you were that voice, and you sat down for 60 seconds and tried to just watch your breath, you would actually be able to. Your mind wouldn't simply wander off. You don't control that voice, you just notice it and think "hey, that's me, I'm thinking this!"
It's not just "thoughts", there's also that little movie that's playing. The one that's repeating the terrible thing that so-and-so said over and over again.
We'll call that voice, that movie, feelings, and everything else in your head, "mental objects". Seeing mental objects and noting that they are not in your control, or even particularly wise, is a core part of understanding Buddhism.
Meditation is the skill, in the "juggling" sense, that you can use to help you intuitively understand the rest of this article.
Emptiness - If I set an apple in front of you, two things appear. First there's the apple, but there's also the mental object of an apple. There's the word "Apple" and a general conception of what an apple is that persists even if the physical apple-stuffs isn't there.
Buddhism says that the actual apple is different than your mental conception of the apple. This is "signlessness" or "emptiness". The "apple stuff" that's over there in the real world is the actual apple, and your mental conception of the apple is an illusion.
If you showed the apple to an alien intelligence, perhaps with entirely different senses, it may perceive the apple as part of the table, it might have no conception of it as food, and it wouldn't call it an "apple".
It's not that the "apple" concept is meaningless, but it's just much more loose and less real than you initially assume. Meditation helps you notice how you brain is assigning concepts to things. You can watch it do it in real time; it happens very quickly, and it changes pretty regularly.
Meditation is a way to watch your brain assign concepts to the world and then cling on to them. You might read a dharma book on emptiness in the same way one might read a book on mushroom foraging. The book explains what mushroom you're trying to find, and so you think "ah, I know about mushroom foraging". But you go to the forest and alas, you find nothing. Gradually, you bumble around the forest until suddenly you see a patch of what you're looking for. And suddenly, you see them everywhere along the path you came from. Now, you actually know about mushroom foraging. It's this second understanding of emptiness that meditation is trying to build in you.
Relative truth - The problem with emptiness as an initial starting point belief is that it makes it really hard to talk about anything.
So Buddhists have a concept of "relative truth." In that, for the purposes of explaining to a lay person, we can say there's an apple on the table. In a "relative truth" the apple is there, but in an "absolute truth" the physical apple on the table is empty of all the "apple" parts (since "appleness" lives in your head, not in reality).
Cause and effect - Buddhists believe in cause and effect and that cause and effect is somewhat "real". It's not just a mental object. Like emptiness, cause and effect is a kind of axiom of Buddhist thought.
Cause and effect is called "Karma", but is not the same "Karma" word that is used in common western speech. Buddhist Karma doesn't mean "justice" or "if you do bad things, bad things will happen to you" or so on. It just means cause-and-effect, without any morality tied to it.
Interbeing - Because cause and effect are real, time is seen as another mental object. The world just is, and our brains just put the "time" concept onto things. We think that the tree and the seed are different, but they are part of the same series of cause and effect, and we're just viewing a cross section.
To help with the terminology, some folks use the word "to inter-be". The seed and the tree "inter-are". Without the seed, the tree would not exist. And if there is a seed, then there's some chance of a tree.
Duhkha - Duhkha is often translated as "suffering", but this translation has been commonly critiqued. It's a subtle word, that I think means "the pain of when your concept for something (form) does not match reality (emptiness)", in that you have the expectation that something should be one way, but it is not actually that way. Other people say "unsatisfactoriness" or "unease".
It's the big-bad of Buddhism and the primary thing it's trying to solve.
It's often said that Buddhism's core solution to problems is to remove "desire" or "lust". However, this always felt like an oversimplification. The solution to Duhkha is not to let-go of the material world, but not to cling to the concepts and labels you're applying to the world.
Impermenance - Initially, I thought of impermanence as "duhka over time", as in the world is changing, and your concepts of the world do not change, and that causes unhappiness. For example, you date the love of your life, and then suddenly you are not dating, and now you are unhappy. I think this is one, very reasonable view of impermanence, that works at one level of relative truth.
However, because of inter-being, I am not convinced the world is changing. The world seems quite fixed, the seed and the tree were always the same. It was the mental formation that changed; the brain chose to think of the seed and the tree differently.
Say you're walking along a trail, and in your fear and anxiety, you see a rattlesnake. However, you look again and it is just a stick blown a bit by the wind. The world didn't change, your conception of the world changed!
What to actually do - All of this sounds fine and made sense to me. However, my misunderstanding for a long time was trying to figure out how to actually apply this in some useful way.
Say you have broken up with the love of your life, your best friend, the most important person to you, and you are devastated. Your mind races with the potential things to do; should you stay in contact? Cut off everything? Say yes to dates with other people? Sit and sob on your floor? You ask everyone you know, and they give you all sorts of things to do, none of which seem satisfactory.
I believe the Buddhist answer is there isn't anything to do.
I thought for awhile this meant the same as the friend saying to "move on", but this is not the case. What would it mean to "move on"? Presumably, it would mean to wrestle your thoughts into suddenly finding the love of your life unlovable, your worst enemy, the least important person, or to shove them out of your brain, forget them entirely. But 1 minute of meditation will show you: the thoughts think themselves. There isn't a "doing" to be done, there isn't a king who can command the world of mental objects to be thus. And if there was such a king, why would you ever command it so? Why would you ever want to see someone you deeply love that way?
I thought for awhile that "letting go" or "not grasping" must, in the heartbreak example, mean letting go of the person. But once again, 1 minute of meditation will show you: the thoughts think themselves. In the mind there is the feeling "this is my person" and you cannot bully the feeling away. It will cling and hold tight. But you can recognize that the feeling is not you, but just another mental object. You can only watch the thought and say kind things as it passes; after all, it lead you to your beloved in the first place. It's wise and has good taste.
But absolute truth is confusing, and so thus, Buddhism has an entire cannon of relative truth suggestions on things to do. I once heard this explained through a story: a student approaches a zen master saying "everything is empty! I can do whatever I want!" and the zen master hits him with a stick and says "was that empty?" So, Buddhism requests that you don't hit people with a stick.
In our heartbreak example, the absolute truth answer is to ask why you were so worried about what to do in the first place. It's a false choice, because you aren't the choice maker, just the observer.
The relative truth is to be kind and minimize the times folks get hit with a stick.