Llama is a family of language models whose weights are published, so they can be run on your own hardware instead of only through an API. It solves the problem that most capable models are reachable only as a hosted service, which rules them out when data cannot leave your network or when per-token cost dominates. It exists because releasing weights turned out to build an ecosystem of fine-tunes and tooling around one architecture, and it is the model most local inference tools target first.
Gemma is a family of open-weight models from Google, published in sizes aimed at running on a single accelerator or a laptop. It solves the access problem for teams that want a capable model they can inspect and run themselves. It exists as the open counterpart to the hosted models, and its significance is that the weights are available for the smaller variants where local inference is realistic.