Drones operating in human-occupied spaces suffer from insufficient communication mechanisms that create uncertainty about their intentions. We present HoverAI, an embodied aerial agent that integrates drone mobility, infrastructure-independent visual projection, and real-time conversational AI into a unified platform. Equipped with a MEMS laser projector, onboard semi-rigid screen, and RGB camera, HoverAI perceives users through vision and voice, responding via lip-synced avatars that adapt appearance to user demographics. The system employs a multimodal pipeline combining VAD, ASR (Whisper), LLM-based intent classification, RAG for dialogue, face analysis for personalization, and voice synthesis (XTTS v2). Evaluation demonstrates high accuracy in command recognition (F1: 0.90), demographic estimation (gender F1: 0.89, age MAE: 5.14 years), and speech transcription (WER: 0.181). By uniting aerial robotics with adaptive conversational AI and self-contained visual output, HoverAI introduces a new class of spatially-aware, socially responsive embodied agents for applications in guidance, assistance, and human-centered interaction.