Units of mass on the atomic scale
The quantity atomic mass (quantity symbol: $m_\mathrm{a}$) is defined as rest mass of a neutral atom in the ground state.
The dimension of the atomic mass is
$$\dim m_\mathrm{a} = \mathsf{M}$$
The coherent SI unit for atomic mass is ‘kilogram’ (unit symbol: $\mathrm{kg}$).  

The quantity relative atomic mass (quantity symbol: $A_\mathrm{r}$) is defined as the ratio of the average mass per atom of an element to 1/12 of the mass of an atom of the nuclide $\ce{^12C}$; i.e. the relative atomic mass is the ratio of the mass of an atom to the unified atomic mass constant:
$$A_\mathrm{r} = m_\mathrm{a}/m_\mathrm{u}$$
where $m_\mathrm{a}$ is the atomic mass and $m_\mathrm{u}$ is the unified atomic mass constant.
The relative atomic mass is a quantity of dimension one (for historical reasons, a quantity of dimension one is often called dimensionless):
$$\dim A_\mathrm{r} = 1$$
The coherent SI unit for relative atomic mass is the unit one (symbol: $1$).
For historical reasons, the IUPAC accepts the use of the special name ‘atomic weight’ for the quantity relative atomic mass. The use of this traditional name is deprecated.

The value of the unified atomic mass constant (symbol: $m_\mathrm{u}$) is defined as 1/12 of the mass of a neutral atom of the nuclide $\ce{^12C}$ in the ground state at rest. 
$$m_\mathrm{u} = \frac{m_\mathrm{a}(\ce{^12C})}{12}$$
The recommended value is
$$m_\mathrm{u} = 1.660\,538\,921(73) \times 10^{-27}\ \mathrm{kg}$$ (source)
Note that the value in SI units is obtained experimentally.

The unit dalton (unit symbol: $\mathrm{Da}$) and the unified atomic
mass unit (unit symbol: $\mathrm{u}$) are non-SI units that are accepted for use with the SI. Actually, ‘dalton’ and ‘unified atomic mass unit’ are alternative names for the same unit, equal to 1/12 times the mass of a free carbon-12 atom, at rest and in its ground state, i.e.

$$1\ \mathrm{Da} = 1\ \mathrm{u} = 1.660\,538\,921(73) \times 10^{-27}\ \mathrm{kg}$$

Thus, the value of the unified atomic mass constant is, by definition, equal to one dalton or one unified atomic mass unit:

$$m_\mathrm{u} = 1\ \mathrm{Da} = 1\ \mathrm{u} = 1.660\,538\,921(73) \times 10^{-27}\ \mathrm{kg}$$

The old relative atomic masses (unfortunately called ‘atomic weights’) and the corresponding atomic mass unit (amu) were originally referred to the relative atomic mass of oxygen, which was taken as 16. However, physicists used the atomic mass of the nuclide $\ce{^16O}$ whereas chemists used the average atomic mass of natural oxygen. This unit became obsolete when IUPAP (1960), IUPAC (1961), ISO, CIPM (1967) and CGPM (1971) agreed to assign the value 12 to the relative atomic mass of the nuclide $\ce{^12C}$.