African languages are spoken by over a billion people, but are underrepresented in NLP research and development. The challenges impeding progress include the limited availability of annotated datasets, as well as a lack of understanding of the settings where current methods are effective. In this paper, we make progress towards solutions for these challenges, focusing on the task of named entity recognition (NER). We create the largest human-annotated NER dataset for 21 African languages, and we study the behaviour of state-of-the-art cross-lingual transfer methods in an Africa-centric setting, demonstrating that the choice of source language significantly affects performance. We show that choosing the best transfer language improves zero-shot F1 scores by an average of 16 points across 21languages compared to using English. Our results highlight the need for benchmark datasets and models that cover typologically-diverse African languages.