Explainable artificial intelligence and interpretable machine learning are research domains growing in importance. Yet, the underlying concepts remain somewhat elusive and lack generally agreed definitions. While recent inspiration from social sciences has refocused the work on needs and expectations of human recipients, the field still misses a concrete conceptualisation. We take steps towards addressing this challenge by reviewing the philosophical and social foundations of human explainability, which we then translate into the technological realm. In particular, we scrutinise the notion of algorithmic black boxes and the spectrum of understanding determined by explanatory processes and explainees' background knowledge. This approach allows us to define explainability as (logical) reasoning applied to transparent insights (into, possibly black-box, predictive systems) interpreted under background knowledge and placed within a specific context -- a process that engenders understanding in a selected group of explainees. We then employ this conceptualisation to revisit strategies for evaluating explainability as well as the much disputed trade-off between transparency and predictive power, including its implications for ante-hoc and post-hoc techniques along with fairness and accountability established by explainability. We furthermore discuss components of the machine learning workflow that may be in need of interpretability, building on a range of ideas from human-centred explainability, with a particular focus on explainees, contrastive statements and explanatory processes. Our discussion reconciles and complements current research to help better navigate open questions -- rather than attempting to address any individual issue -- thus laying a solid foundation for a grounded discussion and future progress of explainable artificial intelligence and interpretable machine learning.