The structure and magnetic properties of orthorhombic $\mathrm{D}{\mathrm{y}}_{2}\mathrm{Ti}{\mathrm{O}}_{5}$ have been investigated using x-ray diffraction, neutron diffraction, and alternating current (ac)/direct current (dc) magnetic susceptibility measurements. We report a continuous structural distortion below 100 K characterized by negative thermal expansion in the [0 1 0] direction. Neutron diffraction and magnetic susceptibility measurements revealed that two-dimensional (2D) magnetic ordering begins at 3.1 K, which is followed by a three-dimensional magnetic transition at 1.7 K. The magnetic structure has been solved through a representational analysis approach and can be indexed with the propagation vector $k=[01/20]$. The spin structure corresponds to a coplanar model of interwoven 2D ``sheets'' extending in the [0 1 0] direction. The local crystal field is different for each $\mathrm{D}{\mathrm{y}}^{3+}$ ion (Dy1 and Dy2), one of which possesses strong uniaxial symmetry indicative of Ising-like magnetic ordering. Consequently, two succeeding transitions under magnetic field are observed in the ac susceptibility, which are associated with flipping each $\mathrm{D}{\mathrm{y}}^{3+}$ spin independently.