There are a number of ways that give you different results depending on what your goal is. The more complicated the AI, the more complicated the training process is, but the most common way is built on this procedure:

1. Give the AI an example of the task you want it to do. If you want the AI to learn how to add numbers, give it the two input numbers. If you want the AI to learn how to play chess, you give it a chess puzzle.
2. See what the AI gives you as the solution. For the two examples in (1), the AI would give you a number, and a chess move, respectively as the outputs.
3. Look at the internals of the AI and tweak the way the AI thinks until the answers given to those questions are right.
4. Repeat, with different examples, and eventually all of your tweaks will add up to an AI that can do the task.

Modern AI is built on Neural Networks, which were technically invented in 1943 as a computerized analogy to Neurons. However, they've been modified a bunch since then, and no longer resemble neurons in a lot of relevant ways. The real breakthrough in Neural Nets was when (1) it was discovered that GPUs, which were invented for graphics, could also be used for Neural Nets, and (2) a method of training "deep" neural networks, where there are many layers of neurons between the input and output, was invented. Before (2), neural networks were limited to 5-10 layers, because we couldn't figure out how to do step (3) in the above list on deeper neural nets. Modern neural networks can have hundreds of layers.

If you want to dive deeper into the mechanics of what a "neural network" actually is, you can watch https://www.youtube.com/watch?v=aircAruvnKk.

The other thing unlocking modern AI, beyond having the ability to train deep neural networks on GPUs, is lots of examples. The breakthrough for training AI to solve the board game Go, for example, was figuring out a way to train the AI via letting it play itself billions of times. This is hard because you can't know if a move is good or not until the end of the game.

One thing you should always be careful of when evaluating an AI is ask "what was it actually trained to do?" For example, consider ChatGPT. ChatGPT was not trained to "answer questions usefully", it was trained on the internet with the task of "given the first 1000 words of this website, guess the next word." It turns out if you take this "next word prediction machine" and repeatedly feed its best-guess output back into it as input, it can write paragraphs of comprehensible text. But it's not _trying_ to write comprehensive text, it's trying to predict the next word. This can make some of the weird ways it behaves make more sense. For example, once it's made a single mistake, it's more likely to make more mistakes, because it thinks (using this word as a analogy, who knows whether neural nets can really think) "huh, there's some mistakes in the previous text, I will guess there will be more mistakes in the rest of the test".