Dark matter was first postulated as a result of Vera Ruben's observations of galaxies' rotation curves, which were flat ($\color{green}{\mathrm{green}}$ in the diagram below), showing that stars revolve much faster than expected based upon Newtonian mechanics ($\color{red}{\mathrm{red}}$ in the diagram below):

Dark matter postulate: Explains why there is much more mass in galaxies than the mass that emits light

Alternative theories of gravity (Modified Newtonian Dynamics (MOND), Weber's gravitational force, etc.): Explains galaxy rotation curves assuming we can "weigh" galaxies accurately based upon the light they emit.

Although it seems Fritz Zwicky can claim priority for having postulated dark matter (see freecharly's answer), Vera Ruben's observations of galaxies' rotation curves gave more evidence to the claim. The rotation curves she observed were flat ($\color{green}{\mathrm{green}}$ in the diagram below), showing that stars revolve much faster than expected based upon Newtonian mechanics ($\color{red}{\mathrm{red}}$ in the diagram below):

Dark matter postulate: Explains why there is much more mass in galaxies than the mass that emits light

Alternative theories of gravity (Modified Newtonian Dynamics (MOND), Weber's gravitational force, etc.): Explains galaxy rotation curves assuming we can "weigh" galaxies accurately based upon the light they emit.

It was first postulated as a result of Vera Ruben's observations of galaxies' rotation curves, which showed stars to be revolving much faster than expected.

Dark matter postulate: Explains why there is much more mass in galaxies than the mass that emits light

Alternative theories of gravity (Modified Newtonian Dynamics (MOND), Weber's gravitational force, etc.): Explains galaxy rotation curves assuming we can "weigh" galaxies accurately based upon the light they emit.

Dark matter was first postulated as a result of Vera Ruben's observations of galaxies' rotation curves, which were flat ($\color{green}{\mathrm{green}}$ in the diagram below), showing that stars revolve much faster than expected based upon Newtonian mechanics ($\color{red}{\mathrm{red}}$ in the diagram below):

Dark matter postulate: Explains why there is much more mass in galaxies than the mass that emits light

Alternative theories of gravity (Modified Newtonian Dynamics (MOND), Weber's gravitational force, etc.): Explains galaxy rotation curves assuming we can "weigh" galaxies accurately based upon the light they emit.