Atoms consist of a nucleus of protons and neutrons surrounded by electrons. Electrons are very light compared to the other two types of particles: a proton is about 1800 times as heavy as an electron.
The electron is one of a class of subatomic particles called leptons which are believed to be fundamental particles (that is, they cannot be broken down into smaller constituent parts). The electron has spin 1/2, which implies it is a fermion, i.e., follows the Fermi-Dirac statistics.
In the Standard Model it forms a doublet in SU(2) with the electron neutrino, as they interact through the weak interaction. The electron has two more massive partners, with the same charge but different masses: the muon and the tau.
When electrons move, free of the nuclei of atoms, and there is a net flow, this flow is called electricity, or an electric current. This might be compared to a flock of sheep moving north together, while the shepherds do not. Electric charge can be directly measured with an electrometer. Electric current can be directly measured with a galvanometer.
So-called "static electricity" is not a flow of electrons at all. More correctly called a "static charge", it refers to a body that has more or fewer electrons than are required to balance the positive charge of the nuclei. When there is an excess of electrons, the object is said to be "negatively charged". When there are fewer electrons than protons, the object is said to be "positively charged". When the number of electrons and the number of protons are equal, the object is said to be electrically "neutral".