Natural language processing (NLP) is a rapidly evolving field at the intersection of linguistics, computer science, and artificial intelligence, which is concerned with developing methods to process and generate language at scale. Modern NLP tools have the potential to support humanitarian action at multiple stages of the humanitarian response cycle. Both internal reports, secondary text data (e.g., social media data, news media articles, or interviews with affected individuals), and external-facing documents like Humanitarian Needs Overviews (HNOs) encode information relevant to monitoring, anticipating, or responding to humanitarian crises. Yet, lack of awareness of the concrete opportunities offered by state-of-the-art techniques, as well as constraints posed by resource scarcity, limit adoption of NLP tools in the humanitarian sector. This paper provides a pragmatically-minded primer to the emerging field of humanitarian NLP, reviewing existing initiatives in the space of humanitarian NLP, highlighting potentially impactful applications of NLP in the humanitarian sector, and describing criteria, challenges, and potential solutions for large-scale adoption. In addition, as one of the main bottlenecks is the lack of data and standards for this domain, we present recent initiatives (the DEEP and HumSet) which are directly aimed at addressing these gaps. With this work, we hope to motivate humanitarians and NLP experts to create long-term impact-driven synergies and to co-develop an ambitious roadmap for the field.