Atomic Number
The atomic number is the number of protons in the nucleus of an atom; it identifies the element and orders the periodic table.
Definition
The atomic number is how many protons an atom has, and each element has its own, like a name tag. Written $Z$, it uniquely identifies the element, equals the electron count in a neutral atom, and orders the periodic table, which is why properties recur periodically. Physically, $Z$ is the nuclear charge in units of $e$; Moseley's 1913 X-ray spectroscopy established it as a fundamental property and fixed the ordering problems in Mendeleev's mass-based table. Together with the neutron number $N$, it gives the mass number $A = Z + N$, and while chemical reactions can never change $Z$, nuclear reactions can.
Example
Hydrogen is number $1$, oxygen is $8$, and gold is $79$. Iron has $Z = 26$, so every iron atom has $26$ protons and a neutral one has $26$ electrons, and the next element, cobalt, has $Z = 27$. In beta-minus decay, carbon-14 ($Z = 6$) becomes nitrogen-14 ($Z = 7$) as a neutron turns into a proton: the mass number stays $14$ but the element changes because $Z$ changed.
Key Insight
Reading the periodic table left to right is just counting protons. Mendeleev ordered elements by mass and misplaced a few; atomic number, not mass, turned out to be the real organizing principle. Every element from $1$ to $118$ (oganesson) has now been observed, so there are no gaps in the sequence.