Glycerol

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Glycerol - a clear, viscous, odourless, slightly sweet, non-toxic, liquid

used e.g. as a humectant, a solvent, a sweetener, a preservative, a thickening agent

has some anti-freeze properties

has a few medical uses

Glycerol and glycerin (US) / glycerine (UK) refer to the same molecule, but glycerin is a commercial term that refers to mixes that are mostly (>95%) glycerol.

You may also see the term glycol. Glycol is a wider group, technically the property/grouping of aliphatic diols, where

aliphatic = non-aromatic

diol = alcohols with two OH functional groups

Glycols include

glycerol (E422)

propylene glycol (E1520 when used for food and E490 when used for cosmetics), somewhat similar to glycerol

ethylene glycol (used in classical antifreeze, and toxic)

Smoke machine fluids were often basically just glycerol (or other glycols, but not ethylene glycol) and water, though there appear to be some newer formulations that leave less residue(verify).

Vaping liquid has a base that is often a mix of part glycerin (smoother, cloudier) and propylene glycol.