"Weights" in LLMs are the "final answer" numbers from the results of model training, and these are the engine of the LLM model. Lets use chocolate chip cookies as an analogy.

The chocolate chip cookies are produced with ingredients, a recipe, labor effort to produce the dough, and then cooking energy/effort to bake chocolate chip cookies. In a traditional AI company, the company gets the ingredients, they write their own recipe, do all the dough creation, and then the energy/labor for baking, and charge you money to get chocolate chip cookies.

An open-weights AI company got all the ingredients, wrote their own recipe, labor effort to produce the dough, and instead of charging you for the dough, you get as much uncooked dough as you want forever. Your only task is to take the dough and cook it yourself and you've got free chocolate chip cookies. You can make as many cookies as you want with your own oven. However, you are not given free raw ingredients, nor are your given the recipe to alter it in a way you might like. You can only get the dough for free.

So open-weight AI models (uncooked dough) are LLMs you can use on your own computers (oven) for free and have LLM output ( chocolate chip cookies), but you don't get the training data (ingredients) nor the training parameters (recipe) that built the open-weight model.

It means the weights (the numeric data that makes up the model) are publicly available to download and use for free.

In some cases there are conditions such as, if you run the model as your business, you need to purchase a license. That's usually the largest, most powerful models, though, not those most people would be able to run at home.

Question, what do you do with it

I use it for both fun throwaway stuff as well as productivity tasks for learning.

Example of fun throwaway:

Do you remember that episode of Seinfeld where the Kramer starts using Facebook marketplace and starts buying the most worthless items before being robbed when trying to get a too-good-to-be-true sale? No? Because it never happened, but you can plug that premise into an crafted LLM prompt and it will pop out a whole TV script with in-character dialog for each actor as well as use of popular existing sets.

Foreign language learning:

I'm studying a foreign language and want to interact with just the level of vocabulary and grammar I have knowledge of right now at my level for practice. I can prompt the LLM to limit itself to just what I know now and adjust the conversation level so I can practice. If I ever get stuck, I can ask the LLM to explain the grammar usage or vocabulary choice.

and how is the data that underlies the model harvested?

I'm not sure what you're asking here. Are you asking, for example, how the Deepseek model was trained? If so, I answered that above. If not, can you rephrase your question?

You run a program like llama.cpp that can use the weights to run the model, which you can then use for whatever you'd use a model for: figuring out tech stuff, coding, etc. It's a bit involved, but there are tools that make it easy to get started. LM Studio for instance.

In some cases, the lab that created the model publishes the dataset that it was trained on, and those are usually made of publicly available data. In most cases, though, the labs don't give details, but the answer likely involves siphoning every web page they could find.