So, to recap the democratic style gives everyone a chance to share their opinions on different matters within the company or the department. The authoritative style, however, gives the responsibility of making decisions to the manager who then delegates tasks to employees in a fair matter. Lastly, the coaching style puts production on the back burner and focuses in on the individual employee and developing their skills.
No matter what style you choose, just be sure that the environment and management style fits the atmosphere of your company. You want your company to be a place people enjoy working at, so keep that in mind!