In the sentence, "Each element of matter has distinct properties and a distinct atomic structure," what does the word structure mean?

Each element has a so-called nucleus which possesses all the positive charge and MOST of the mass of the atom. The number of positively charged particles in the nucleus, #Z#, is the so-called atomic number. #Z# determines the identity of the element: #Z=1#, hydrogen; #Z=2#, helium; #Z=3#, lithium; #Z=43#, cadmium.

Neutrons, massive NEUTRALLY charged nuclear particles also reside in the nucleus. (In other words the nucleus consists of these massive particles). Different numbers of neutrons in the nucleus give rise to the existence of isotopes, i.e. #""^1H#, protium, #""^2H#, deuterium, #""^3H#, tritium, are all hydrogen isotopes (because #Z# #=# #1#), but they have different masses because they have different numbers of NEUTRONS in their nuclei.

Electrons, fundamental negatively charged particles of negligible mass, are conceived to orbit around the nucleus in strict geometries (defined by a wave function) and with quantized levels of energy (quantum numbers define these functions and energy). In the NEUTRAL atom, the number of electrons is equal to #Z#, the atomic number. The wave function, its shape, and it energy, relates to electronic structure.

Nuclear structure (including the strong nuclear force which binds the nucleus together) is outside the ambit of undergraduate chemistry.