Every choice and option has to do with the level of management and control you (as the IT) and your company wants to have. If your goal is just to share the internet and maybe a few printers/files, there is no real need for AD. If on the other hand you want to have user credentials (who logs in and when/where), application restrictions (who is using what), shared pools of disk and quota per user and applications, Distributed applications, Server Pools, Virtual Storage/ Radius Internet access, etc, then Active directory or alternatives are a one-way street.
There are alternatives mainly Samba, Jumpcloud, etc (google is your friend) but I will be honest... Active directory just works unless you have the time to invest (and a lot depends on the kind of services/apps you need for your network). Alternatively, you can move everyone to the cloud if the needs are office apps/ file-sharing / collaboration/communication only.
In any case, yes you understood right, there must be a server just for the AD management (depends on No of users and No of services if the server is dedicated or not) called domain controller but the whole thing needs very careful planning cause once you have set the server/client environment, any downtime on the server for updates/changes/mistakes will be downtime for the company. You need to have a backup (both software/hardware) and know a bit about how to secure it otherwise... Keep in mind there might be changes/additions you may also have to do on the other PCs in order to join the domain.
As you see tho, your question may be simple but the answer isn't, and as we will dig further in, it will become very difficult or time-consuming to explain the whats and hows, and everything we will talk about, will probably generate more questions than it answeres. The process needs good design and people who know how to administer and secure a network both in theory and practice, and it's not something simple, especially for someone with zero experience. Dont get me wrong, it's not as hard as I may make it sound, with some dedication and time, I'm sure you can make it work... but it needs a lot of experimentation and tests something which you only know if there are time/resources to do it.
You can find some good guides on the net both in book or video forms or even you may want to talk to a consultant to explore options but either way, plan ahead very well before you start.
