From Cohen-Tannoudji (Cohen-Tannoudji 1977):
“Two particles are said to be identical if all their intrinsic properties (mass, spin, charge, etc.) are exactly the same: no experiment can distinguish one from another. Thus, all electrons in the universe are identical, as are all the protons and all the hydrogen atoms[...]. An important consequence can be deduced from this definition: when a physical system contains two identical particles, there is no change in its properties or its evolution if the roles of these two particles are exchanged.”
In classical physics it is possible to keep track of individual particles even though they may look alike. If we have a system consisting of two particles A and B, we can, in principle, follow the trajectory of 1 and that of 2 separately at each instant of time. In quantum mechanics, however, identical particles are truly indistinguishable. This is because we cannot specify more than a complete set of commuting observables for each of the particles. Nor can we follow the trajectory because it would entail a position measurement at each instant of time (Sakurai 1993).