With the ever-increasing computing power of supercomputers and the growing scale of scientific applications, the efficiency of MPI collective communications turns out to be a critical bottleneck in large-scale distributed and parallel processing. The large message size in MPI collectives is particularly concerning because it can significantly degrade the overall parallel performance. To address this issue, prior research simply applies the off-the-shelf fix-rate lossy compressors in the MPI collectives, leading to suboptimal performance, limited generalizability, and unbounded errors. In this paper, we propose a novel solution, called C-Coll, which leverages error-bounded lossy compression to significantly reduce the message size, resulting in a substantial reduction in communication cost. The key contributions are three-fold. (1) We develop two general, optimized lossy-compression-based frameworks for both types of MPI collectives (collective data movement as well as collective computation), based on their particular characteristics. Our framework not only reduces communication cost but also preserves data accuracy. (2) We customize SZx, an ultra-fast error-bounded lossy compressor, to meet the specific needs of collective communication. (3) We integrate C-Coll into multiple collectives, such as MPI_Allreduce, MPI_Scatter, and MPI_Bcast, and perform a comprehensive evaluation based on real-world scientific datasets. Experiments show that our solution outperforms the original MPI collectives as well as multiple baselines and related efforts by 1.8-2.7X.