Deep speaker embeddings have shown promising results in speaker recognition, as well as in other speaker-related tasks. However, some issues are still under explored, for instance, the information encoded in these representations and their influence on downstream tasks. Four deep speaker embeddings are studied in this paper, namely, d-vector, x-vector, ResNetSE-34 and ECAPA-TDNN. Inspired by human voice mechanisms, we explored possibly encoded information from perspectives of identity, contents and channels; Based on this, experiments were conducted on three categories of speaker-related tasks to further explore impacts of different deep embeddings, including discriminative tasks (speaker verification and diarization), guiding tasks (target speaker detection and extraction) and regulating tasks (multi-speaker text-to-speech). Results show that all deep embeddings encoded channel and content information in addition to speaker identity, but the extent could vary and their performance on speaker-related tasks can be tremendously different: ECAPA-TDNN is dominant in discriminative tasks, and d-vector leads the guiding tasks, while regulating task is less sensitive to the choice of speaker representations. These may benefit future research utilizing speaker embeddings.