These are models that are used in NLP to understand and/or output human language. There are two main kinds: statistical and AI models.
The best way to describe language models is a "fancy autocomplete" (actually, autocomplete uses language models under the hood to do its thing, but for the sake of analogy just pretend it doesn't). It simply looks at incoming text input, and tries to predict the next word.
Article with full explanation: https://www.geeksforgeeks.org/what-are-language-models-in-nlp/
A language model that has a very large number of parameters (often in the hundreds of billions), and has been trained on an extremely large dataset, so that it has a broad knowledge of many topics.
Video explaining large language models, from a mathematical perspective: