Interpretability in recommender systems is pivotal for fostering user trust and comprehension in decision-making processes. This paper offers a comprehensive synthesis of existing research and classification frameworks on interpretability methods within recommender systems. We begin by reviewing various interpretability approaches across diverse domains and applications, organizing existing classifications based on underlying principles, objectives, and applicability. Subsequently, we focus on explainable recommender systems (XRS), highlighting the importance of transparency and interpretability in recommendation processes. Leveraging insights from prior literature, we present a synthesized overview tailored specifically to XRS, categorizing approaches based on interpretability mechanisms, granularity of explanations, and user interaction paradigms. Our synthesis aims to equip researchers and practitioners with a structured foundation for understanding, comparing, and advancing interpretable recommendation algorithms. By prioritizing interpretability, we empower users, mitigate biases, and enhance overall user experiences in digital platforms. Moving forward, further refinement of these methods is crucial to align with ethical principles and societal values, ensuring trust and transparency in recommendation systems. This paper contributes to the ongoing discourse surrounding the ethical, social, and practical implications of recommendation systems in contemporary society.