Atom
An atom is the smallest unit of an element that keeps its chemical identity, built from protons, neutrons, and electrons.
Definition
An atom is the smallest piece of an element that is still that element, far too small to see with your eyes. It has a dense central nucleus of protons and neutrons surrounded by electrons, and the number of protons decides which element it is; when electrons equal protons the atom is electrically neutral. Atoms are on the order of $10^{-10}\text{ m}$ across (about one angstrom), while the nucleus is roughly $10^{-15}\text{ m}$, so the atom is almost entirely empty space filled by electrons in quantized orbitals. An atom's chemistry is governed by its electron configuration, especially the valence electrons in the outermost shell.
Example
A gold ring is trillions of gold atoms; cut it smaller and smaller and the last piece that is still gold is one atom. A carbon atom has $6$ protons, usually $6$ neutrons, and $6$ electrons, while oxygen has $8$ protons and $8$ electrons, and two hydrogens bonded to one oxygen make $H_2O$. Sodium (atomic number $11$) has the configuration $1s^2 2s^2 2p^6 3s^1$, and that single loosely held $3s$ electron is why sodium reacts violently with water and forms $Na^+$ in salts like $NaCl$.
Key Insight
If an atom were the size of a football stadium, its nucleus would be about a marble at midfield. Chemical reactions only rearrange electrons, so no chemical process can turn one element into another; that takes a nuclear reaction that changes the proton count. Our picture of the atom moved from Dalton's solid ball to Thomson's electron, Rutherford's nucleus, Bohr's energy levels, and the quantum mechanical model, each kept only as long as it explained the evidence.