The ever-growing number of people using Twitter makes it a valuable source of timely information. However, detecting events in Twitter is a difficult task, because tweets that report interesting events are overwhelmed by a large volume of tweets on unrelated topics. Existing methods focus on the textual content of tweets and ignore the social aspect of Twitter. In this paper we propose MABED (i.e. mention-anomaly-based event detection), a novel statistical method that relies solely on tweets and leverages the creation frequency of dynamic links (i.e. mentions) that users insert in tweets to detect significant events and estimate the magnitude of their impact over the crowd. MABED also differs from the literature in that it dynamically estimates the period of time during which each event is discussed, rather than assuming a predefined fixed duration for all events. The experiments we conducted on both English and French Twitter data show that the mention-anomaly-based approach leads to more accurate event detection and improved robustness in presence of noisy Twitter content. Qualitatively speaking, we find that MABED helps with the interpretation of detected events by providing clear textual descriptions and precise temporal descriptions. We also show how MABED can help understanding users' interest. Furthermore, we describe three visualizations designed to favor an efficient exploration of the detected events.