High critical-temperature (Tc) superconductivity in the cuprates has been a defining challenge of condensed matter physics for the 35 years since their discovery. One strategy to address this challenge has been to look for "cuprate analog'' materials: alternative transition metal oxides that exhibit ingredients that are considered proxies for cuprate physics. These key ingredients include a quasi-2D structure based on the CuO2 planes, a nominal oxidation state for Cu2+: 3d9 with a single hole in the uppermost dx2-y2 orbital, and a strong O(2p) and Cu(3d) hybridization. Nickelates have been an obvious choice of study in this context due to the proximity of Ni to Cu on the periodic table. After a 30 year wait, superconductivity in nickelates was realized for the first time in 2019 in hole-doped NdNiO2 (Li et al, 2019). This material contains NiO2 planes (analog to the CuO2 planes of the cuprates), and realizes a Ni1+ oxidation state (analog to Cu2+). NdNiO2 is simply the infinite-layer member of a larger family of materials represented by the chemical formula Rn+1NinO2n+2 (R= La, Pr, Nd; n >= 2), where n refers to the number of NiO2 planes along the c axis. In this thesis, a comprehensive description of the electronic structure of the Rn+1NinO2n+2 family of layered rare-earth nickelates (for n= oo and n=2-6) using state-of-the-art first-principles methods is presented. Specifically, different levels of theory are used to describe the electronic structure of this family of materials: from density-functional band theory (DFT) to incorporating correlation effects at the mean-field level via DFT+U, and finally including dynamical many-body effects via DFT+dynamical mean-field theory (DFT+DMFT). It is shown that the cuprate-like character of the layered nickelate series increases from the n=oo to the n=3 members. Namely, as n decreases the electronic structure becomes more single-band-like, and the degree of p-d hybridization increases while correlations are dominated by the dx2-y2 orbitals. Insights from these calculations allowed for the prediction of the n=4-6 nickelates as ideal candidates for nickelate superconductivity. Indeed, superconductivity was subsequently observed in the quintuple layer nickelate Nd6Ni5O12 (Pan et al, 2021). That superconductivity arises in this layered rare-earth nickelate series, suggests that a new family of superconductors has been uncovered, currently with two members, n=oo and n=5.