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As the emerging trend of graph-based deep learning, Graph Neural Networks (GNNs) excel for their capability to generate high-quality node feature vectors (embeddings). However, the existing one-size-fits-all GNN implementations are insufficient to catch up with the evolving GNN architectures, the ever-increasing graph sizes, and the diverse node embedding dimensionalities. To this end, we propose \textbf{GNNAdvisor}, an adaptive and efficient runtime system to accelerate various GNN workloads on GPU platforms. First, GNNAdvisor explores and identifies several performance-relevant features from both the GNN model and the input graph, and uses them as a new driving force for GNN acceleration. Second, GNNAdvisor implements a novel and highly-efficient 2D workload management, tailored for GNN computation to improve GPU utilization and performance under different application settings. Third, GNNAdvisor capitalizes on the GPU memory hierarchy for acceleration by gracefully coordinating the execution of GNNs according to the characteristics of the GPU memory structure and GNN workloads. Furthermore, to enable automatic runtime optimization, GNNAdvisor incorporates a lightweight analytical model for an effective design parameter search. Extensive experiments show that GNNAdvisor outperforms the state-of-the-art GNN computing frameworks, such as Deep Graph Library ($3.02\times$ faster on average) and NeuGraph (up to $4.10\times$ faster), on mainstream GNN architectures across various datasets.