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Here we report the first principles study of yttrium-doped Ba(Zr$_x$Ti$_{1-x}$)O$_3$ (Y-doped BZT) for $x=0.125$, $0.250$, and $0.375$ supercells as a promising rare earth doped lead-free ferroelectric material. The local and electronic structures of Y-doped BZT and pure BZT relaxed supercells are systematically investigated in term of atomic pair distribution functions (PDFs), A and B site cation off-centering, and electronic density of states (DOS) respectively. The PDF spectrums show the increasing of structural disorder as a function of Zr concentration, while the short-range disorder is strongly influenced by the size of cation and their corresponding vacancies. Moreover, the drastic diffusion in PDF spectrums of Y-doped BZT reflects the compositional fluctuation on the local environment of the supercells. For cation off-centering, Ti$^{4+}$ play a major role as the active ferroelectric distortion in BZT. However, the substitution of yttrium ions into BZT matrix enhanced the lattice distortion as observed from the increasing of off-centering magnitude for both A and B site cations. The calculated total and projected DOSs on $d$ states of cations and O-$2p$ states confirm the strong hybridization between Ti $3d$ and O $2p$ states, thus implying the covalent bonding in Ti-O$_6$ octahedral. On the other hand, the little effects of yttrium ions on the electronic structures, especially the highest and lowest valence and conduction bands, were observed.