When people in AI say "the model", they mean the core system that was trained on data and now produces answers. Not the website, not the app, not the chat window. Those are wrappers around the model (see harness).
Models come in families and sizes. A company like Anthropic or OpenAI offers larger models that are more capable and cost more, and smaller ones that are faster and cheaper. The most capable tier at any moment is called a frontier model. Text models are called LLMs, and models that also handle images, audio or video are called multimodal.
Why you keep hearing it
Because the model is the unit of progress in AI. New releases ("a new model dropped") are what change what is possible. Product announcements name the model they run on. And people compare tools by asking which model sits underneath.
What it means for you
Two takeaways. First, when a tool underwhelms you, check which model it uses; an old or small model explains a lot. Second, remember that many competing products run on the same few models. The differences you feel usually come from everything built around the model, not the model itself.