The standard toothpaste abrasive and thickener. It scrubs off plaque and surface stain; how harsh a paste is depends on the grade, which labels rarely show.
Hydrated silica is a synthetic, non-crystalline form of silica ground to fine particles. Grades differ: some mainly thicken the paste, others are cleaning abrasives. It is compatible with every type of fluoride, which is why it is the most common abrasive.
Abrasiveness is measured as RDA (relative dentin abrasivity). International standards cap toothpaste at 250; most everyday pastes sit well below that, while 'whitening' pastes are often higher. It is not the crystalline silica dust linked to lung disease, and there is no inhalation risk in a paste.
Abrasives are what let toothpaste remove extrinsic stain and help brushing clear plaque, shown in lab and short clinical stain studies. Much of the literature comes from manufacturers. Whether a particular paste's abrasivity wears exposed root surfaces over years is not well studied in people.
The abrasive system is the key functional ingredient of whitening toothpastes, designed to remove extrinsic stain without undue wear; peroxides, enzymes, pyrophosphates or optical agents may be added. Clinical tests typically run 2-6 weeks and measure surface stain.