BusyBox gives you the core tools and commands of Linux, but it is not itself a terminal. There are several good terminal apps in the Play Store though. Typically, you want to install both BusyBox and also a terminal app.
You can certainly use it to do a lot of things without root. Of course you won't be able to type su and access root commands if your phone is not rooted.
As mentioned, if you want access to busybox libraries, you can install that without root with an app like this
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