The modern atomic model involves a dense atomic nucleus containing a fixed number of protons and neutrons surrounded by a probabilistic cloud of electrons. This model was established following the discovery of quantum mechanical properties such as wave-particle duality and uncertainty.
Unlike the more common Bohr depiction of the atom that involves point-particle electrons orbiting the nucleus in fixed, circular paths, electrons in modern atomic theory exist as waveforms surrounding the nucleus.These electron waveforms exhibit both particle and wave properties and are described by the Schrodinger wave equation. The uncertainty principle of quantum mechanics makes it impossible to determine exactly where an electron is around the atom; so instead, volumes that have a high probability of having electrons are defined. The probability in these volumes ranges between 0's at their edges, indicating the complete absence of electrons, trending to, but never reaching, 1, indicating an absolute certainty of the presence of an electron.
These volumes are called orbitals and take different shapes depending on the energy level and sublevel of the electrons. There are four kinds of orbitals, designated s, p, d and f. The s orbital is the simplest, consisting of a simple spherical shell of electrons. The p orbital is more complex, consisting of a double-bowtie shape that extends in three directions that are perpendicular to one another. The d and f orbitals are even more complex in shape, and are usually omitted from textbook discussions.