Quantum many-body scars (QMBS) provide a route to weak ergodicity breaking in isolated many-body systems. Unlike conventional mechanisms for the failure of thermalization, such as integrability or many-body localization, where atypical behavior pervades the entire spectrum, QMBS systems harbor only a small, measure-zero set of nonthermal eigenstates embedded within an otherwise thermalizing bulk. Conventionally, these atypical states ($scar$ states) are distinguished by their anomalously low entanglement entropy relative to nearby thermal eigenstates. When a system is initialized in a state with significant overlap onto this scarred subspace, its subsequent dynamics evade rapid thermalization, instead producing long-lived coherent oscillations in local observables. QMBS have now been identified across a diverse range of platforms, including lattice models, Floquet systems, and various quantum simulators. Nevertheless, a systematic microscopic understanding of how scar eigenstates arise and how they can be analytically constructed remains incomplete. The central aim of this dissertation is to use interacting spin models as a concrete setting to investigate the explicit construction, algebraic organization, and entanglement structure of a hierarchy of QMBS, including states that lie well beyond the low-entanglement paradigm. The first part of this thesis studies the spin-$1$ Kitaev chain, where conserved bond parity operators fragment the Hilbert space into exponentially many dynamically disconnected sectors. Beyond the previously identified sector that maps exactly onto the paradigmatic PXP model, we show that pronounced scarred dynamics also occur in other sectors. Suitably chosen product states in these sectors display strong and long-lived fidelity revivals, which we analyze using the forward-scattering approximation, revealing how constrained Hilbert spaces give rise to robust non-ergodic dynamics. We then develop a general algebraic construction of exact zero-energy eigenstates with volume-law entanglement in a large class of spin chains with periodic boundary conditions. These states are formed by entangled dimers on antipodal sites and arise in the middle of the spectrum for models including the transverse-field Ising model, the PXP model, and spin-$S$ $XY$ and Kitaev chains. Although they exhibit thermal expectation values for all local observables, they remain atypical under suitable nonlocal few-body probes, thereby extending the notion of scar-like eigenstates decisively beyond the low-entanglement paradigm. The construction generalizes to spin models on arbitrary graphs in higher dimensions. Finally, we revisit the spin-$1$ $XY$ chain and construct new families of exact scars within its extensively degenerate zero-energy manifolds, arising from the interplay between $U(1)$ magnetization conservation and chiral symmetries. These include interference-protected Fock-space-cage-like states, volume-entangled towers, and mirror-dimer states. Using the commutant algebra framework, we show that these states are simultaneous eigenstates of non-commuting local operators, providing a unifying algebraic organizing principle and systematic routes to identifying and classifying QMBS in generic many-body systems.