Aloe Barbadensis
The suffix tells you what part was used and how it was processed. "Leaf juice" generally refers to liquid from the inner leaf gel, often dried to a powder and reconstituted with water during manufacturing. "Leaf extract" is a more concentrated preparation carried in a solvent, and "leaf water" is a diluted form. Where the name sits in the ingredient list gives a rough sense of how much is present.
What it contributes is mainly water, polysaccharides and sugars, giving a light, cooling, slightly humectant feel — which is why it often appears near the top of gel moisturisers, sheet masks and after-sun products.
Two practical notes. The processed cosmetic ingredient is not the same thing as sap scraped from a houseplant, which can carry the yellow latex layer containing aloin — an irritant — and has no preservation, so it spoils quickly. And although allergy to aloe is uncommon, it does happen, so patch test before spreading a new product over a large area.