Learn Russian: Мой дядя заведует семейным бизнесом и финансами. - My uncle manages the family business and finances.

In English, you manage something, which feels like a direct object. However, Russian verbs of control, leadership, and possession (like заведовать) require the Instrumental case for the thing being managed. You can think of it historically as directing by means of something. Both бизнесом (masculine singular) and финансами (plural) take the Instrumental endings here because they are governed by the verb.

What is the difference between заведовать and управлять if they both mean to manage?

Both verbs take the Instrumental case, but they have different flavors. Заведовать usually means to be the official head of a specific department, facility, or operation (like being the manager of a store, a clinic, or a budget). Управлять is a bit broader; it can mean managing an abstract process, steering a company, or even physically operating a vehicle (like driving a car).

Why does the -ова- in the infinitive заведовать disappear in the form заведует?

This is a standard conjugation pattern for verbs ending in -овать or -евать that you likely learned earlier but is easy to forget. In the present and simple future tenses, the -ова- or -ева- suffix changes to -у- or -ю-. Therefore, заведовать becomes я заведую, ты заведуешь, and for the uncle (he), он заведует.