Recently, various neural models for multi-party conversation (MPC) have achieved impressive improvements on a variety of tasks such as addressee recognition, speaker identification and response prediction. However, these existing methods on MPC usually represent interlocutors and utterances individually and ignore the inherent complicated structure in MPC which may provide crucial interlocutor and utterance semantics and would enhance the conversation understanding process. To this end, we present MPC-BERT, a pre-trained model for MPC understanding that considers learning who says what to whom in a unified model with several elaborated self-supervised tasks. Particularly, these tasks can be generally categorized into (1) interlocutor structure modeling including reply-to utterance recognition, identical speaker searching and pointer consistency distinction, and (2) utterance semantics modeling including masked shared utterance restoration and shared node detection. We evaluate MPC-BERT on three downstream tasks including addressee recognition, speaker identification and response selection. Experimental results show that MPC-BERT outperforms previous methods by large margins and achieves new state-of-the-art performance on all three downstream tasks at two benchmarks.