  Since 2008, many new families of iron-based high temperature (high-$T_c$)
superconductors have been discovered \cite{Hosono,ChenXH,FeTe,ChenXL}. Unlike
all parent compounds of cuprates that share a common antiferromagnetically (AF)
ordered ground state, those of iron-based superconductors exhibit many
different AF ordered ground states, including collinear-AF (CAF) state in
ferropnictides \cite{caf}, bicollinear-AF (BCAF) state in 11-ferrochalcogenide
$FeTe$ \cite{bcaf,bcaf2}, and block-AF (BAF) state in 122-ferrochalcogenide
$K_{0.8}Fe_{1.6}Se_2$ \cite{baf}. While the universal presence of
antiferromagnetism suggests that superconductivity is strongly interrelated
with magnetism, the diversity of the AF ordered states obscures their
interplay. Here we show that all magnetic phases can be unified within an
effective magnetic model. This model captures three incommensurate magnetic
phases, two of which have been observed experimentally. The model characterizes
the nature of phase transitions between the different magnetic phases and
explains a variety of magnetic properties, such as spin-wave spectra and
electronic nematism. Most importantly, by unifying the understanding of
magnetism, we cast new insight on the key ingredients of magnetic interactions
which are critical to the occurrence of superconductivity.
