In this master thesis we study affine toric varieties in a complex space as a set of affine algebraic varieties, that can be defined using combinatorial data and are therefore to some extent easier to work with in comparison to general affine varieties. Their basic properties are established, as well as their construction from polyhedral cones of a real vector space. We introduce deformation space $T^{1}_{X},$ which is closely studied in a special case, when polyhedral cones are constructed from embedding of lattice polygon to $n$--th level (regarding the last parameter) of this space. In this case, homogenous components of space $T^{1}_{X}$ have an important geometric interpretation. We focus on a difference in research if a polygon is embedded on the first or on the higher level of a real space, as this strongly impacts the complexity of the deformation space itself. At the end, a monoid $\tilde{T}$ is described as a further important object of the deformation theory of affine toric varieties.
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