The relative atomic mass of sodium is not 23 amu (amu = atomic mass unit), it is 23 with no units (as is suggested by the word "relative").

The relative atomic mass is the ratio of the average mass of an atom of an element (given in amu) to 1 amu. For atoms, the atomic mass one typically uses is essentially the sum of the masses of the protons and neutrons which comprise the nucleus of the atom; the mass of each electron (~1/2000 amu) is negligible in comparison. Protons and neutrons have similar masses, approximately 1 amu each. Most sodium atoms have 11 protons and 12 neutrons, giving a total of ~23 amu, and dividing by 1 amu then gives 23 for the relative atomic mass. Note the use of most in the preceding sentence. The number of protons determines the chemical identity of an element, but the number of neutrons can vary - these are different isotopes of an element.

The standard atomic weight takes into account the different relative abundances and masses of the isotopes by weighting the mass of isotopes by the frequency of their occurrence. This is why some elements have fractional standard weights (e.g. copper at 63.55).

It's because sodium atoms each have 11 protons, with a mass of 1, and 12 neutrons, with a mass of 1.

Number of Protons/Electrons‎: ‎11
Number of Energy Levels‎: ‎3
Number of Neutrons‎: ‎12
Atomic Mass‎: ‎22.98977 amu

Most sodium atoms contain twelve neutrons. That is the most common isotope of sodium. Because each neutron weighs one AMU, and because each of sodium's eleven protons also weighs one AMU, twelve plus eleven is twenty-three.

It's fairly complicated nuclear physics to understand why some isotopes are stable and some aren't, but it is a general trend that the more protons you have, the more neutrons you need, and the number of neutrons needed goes up faster than the number of protons.

Relative atomic mass is calculated by dividing the atomic mass of an element by what we call the "unified atomic mass unit".

The unified atomic mass unit is defined as 1/12 of the mass of a carbon atom with 6 protons, 6 neutrons, and 6 electrons. The relative atomic mass of sodium is calculated by dividing the atomic mass of sodium by the unified atomic mass unit, which is 1.66*10^-27 kg. The number of 23 amu means that a normal, or stable, sodium atom weighs 23 unified atomic mass units. Since sodium atoms have 11 protons, this means that a normal sodium atom has 23 - 11 = 12 neutrons. It would also have 11 electrons, but electrons weigh so little compared to protons and neutrons that the electron masses can be approximated to 0. Note that by "stable", we mean that the sodium atom will not very quickly change into another element by losing parts of its nucleus.