Phishing is one of the most prevalent and expensive types of cybercrime faced by organizations and individuals worldwide. Most prior research has focused on various technical features and traditional representations of text to characterize phishing emails. There is a significant knowledge gap about the qualitative traits embedded in them, which could be useful in a range of phishing mitigation tasks. In this paper, we dissect the structure of phishing emails to gain a better understanding of the factors that influence human decision-making when assessing suspicious emails and identify a novel set of descriptive features. For this, we employ an iterative qualitative coding approach to identify features that are descriptive of the emails. We developed the ``Phishing Codebook'', a structured framework to systematically extract key information from phishing emails, and we apply this codebook to a publicly available dataset of 503 phishing emails collected between 2015 and 2021. We present key observations and challenges related to phishing attacks delivered indirectly through legitimate services, the challenge of recurring and long-lasting scams, and the variations within campaigns used by attackers to bypass rule-based filters. Furthermore, we provide two use cases to show how the Phishing Codebook is useful in identifying similar phishing emails and in creating well-tailored responses to end-users. We share the Phishing Codebook and the annotated benchmark dataset to help researchers have a better understanding of phishing emails.