Accurate segmentation of multiple abdominal organs from Computed Tomography (CT) images plays an important role in computer-aided diagnosis, treatment planning and follow-up. Currently, 3D Convolution Neural Networks (CNN) have achieved promising performance for automatic medical image segmentation tasks. However, most existing 3D CNNs have a large set of parameters and huge floating point operations (FLOPs), and 3D CT volumes have a large size, leading to high computational cost, which limits their clinical application. To tackle this issue, we propose a novel framework based on lightweight network and Knowledge Distillation (KD) for delineating multiple organs from 3D CT volumes. We first propose a novel lightweight medical image segmentation network named LCOV-Net for reducing the model size and then introduce two knowledge distillation modules (i.e., Class-Affinity KD and Multi-Scale KD) to effectively distill the knowledge from a heavy-weight teacher model to improve LCOV-Net's segmentation accuracy. Experiments on two public abdominal CT datasets for multiple organ segmentation showed that: 1) Our LCOV-Net outperformed existing lightweight 3D segmentation models in both computational cost and accuracy; 2) The proposed KD strategy effectively improved the performance of the lightweight network, and it outperformed existing KD methods; 3) Combining the proposed LCOV-Net and KD strategy, our framework achieved better performance than the state-of-the-art 3D nnU-Net with only one-fifth parameters. The code is available at https://github.com/HiLab-git/LCOVNet-and-KD.