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I recently read up on some chemistry and found these units of mass:
Are they all the same unit but just with different names or are they different? I'm looking for a simple explanation as I'm new to chemistry.
It is a simple question that requires a rather complicated answer.
Origins
For technical reasons, in 1898, chemist Wilhelm Ostwald and others proposed to redefine the unit of atomic mass as 1/16 of the mass of an oxygen atom. That proposal was formally adopted by the International Committee on Atomic Weights (ICAW) in 1903. ... This suggestion was made before the discovery of the existence of elemental isotopes, which occurred in 1912. ... This definition remained unchanged until 1961.$^1$
I think the Wikipedia article is wrong about the name. I think the notion was initially conceived as "atomic weight," not "atomic mass."
atomic mass unit(amu)
The discovery of isotopes of oxygen in 1929 required a more precise definition of the unit. Unfortunately, two distinct definitions came into use. Chemists choose to define the amu as 1/16 of the average mass of an oxygen atom as found in nature; that is, the average of the masses of the known isotopes, weighted by their natural abundance. Physicists, on the other hand, defined it as 1/16 of the mass of an atom of the isotope oxygen-16 ($\ce{^{16}O}$).$^1$
unified atomic mass unit(u)
...in 1961 the International Union of Pure and Applied Chemistry (IUPAC), which had absorbed the ICAW, adopted a new definition of the atomic mass unit for use in both physics and chemistry; namely, 1/12 of the mass of a carbon-12 atom. ... The new unit was named the "unified atomic mass unit" and given a new symbol "u", to replace the old "amu" that had been used for the oxygen-based units. However, the old symbol "amu" has sometimes been used, after 1961, to refer to the new unit, particularly in lay and preparatory contexts.$^1$
I'll be less diplomatic. Since 1961 "atomic mass unit" has been conflated with the "unified atomic mass unit."
dalton
In 1993, the IUPAC proposed the shorter name "dalton" (with symbol "Da") for the unified atomic mass unit.$^1$
"And now..., for the rest of the story" as Paul Harvey used to say.
Until the 2019 redefinitions of SI units, the number of "entities" in a mole was determined experimentally. Since 1889 the kilogram was the mass of a particular platinum/iridium cylinder. With the 1961 definition of the unified atomic mass unit, exactly 12 grams of $\ce{^{12}C}$ had exactly 1 mole of carbon atoms. So the experimental measure of a mole was based on how many $\ce{^{12}C}$ atoms would be in exactly 12/1000 of a kilogram.
However in 2019 both the kilogram and the mole were redefined to give them particular values that would no longer depend on experimental measurements. This was in a continuance of an effort by scientists to define a set of primary physical constants that didn't depend on particular physical standards such as scratches on a bar to determine the meter or a blob of metal to determine the kilogram.
The point of this is that with the mole and kilogram now being given definite fixed values, exactly one mole of carbon-12 atoms no longer has a mass of exactly 12 grams by definition. Thus now $u≈\pu{1.66053906660(50)\times 10^{−27} kg}$.
tl;dr For practical purposes the three terms are synonymous as user Maurice noted. It would seem that the preferred term is now the dalton.
$^1$ Wikipedia's article on the dalton (Da).
$^2$ Wikipedia's article on the kilogram.
