Bronsted acids are proton donors and must have a ready supply of reactive hydrogen. But Lewis acids can react to make a solution acidic even without having such a supply within the molecule.
Aluminum chloride is a good example of this sort of acid. Aluminum chloride cannot donate protons by itself. But acting as a Lewis acid it coordinates with electron pairs on the oxygen atoms of water molecules and thus induces the water to transfer protons to the rest of the solvent. The Lewis acid aluminum chloride is not a Bronsted acid but can make other things, in this case water, become Bronsted acids.
That's why chemists use the same word, "acid", to describe the seemingly very different processes of proton donation and electron-pair acceptance.