<p>Quantum computing, leveraging quantum phenomena like superposition and entanglement, is emerging as a transformative force in computing technology, promising unparalleled computational speed and efficiency crucial for engineering applications. This paper provides a domain-specific review tailored to computational mechanics, contextualizing quantum algorithms within the language and challenges of scientific computing—particularly for problems such as linear systems, ODEs, PDEs, and Hamiltonian simulations. We synthesize key algorithmic building blocks, including block encoding, qubitization, quantum signal processing, and amplitude amplification, and demonstrate their relevance through practical examples and annotated Qiskit code. The review uniquely bridges foundational theory and implementation by surveying the full quantum software stack, from hardware abstractions to circuit-level design. In doing so, we lower the barrier to entry for engineering researchers and identify key obstacles to adoption, such as the construction of domain-specific oracles and the current limitations of Noisy Intermediate-Scale Quantum (NISQ) hardware for large-scale mechanics problems. This paper aims to serve as both a primer and a roadmap for advancing the integration of quantum computing into computational mechanics.</p>