anrufen is to call someone with the telephone.
rufen is to call someone in the sense that you contact them so that they come. It can be on the telephone, or it can mean that you yell loudly, for example. Usually it's used in the sense of "calling the police" (die Polizei rufen) - you're contacting them that they come to you, not to have a chat. In the old days, maybe you could also "call a doctor" who would then visit your house.
There may be other shades of meaning for both words based on context, but that's the main distinguishing feature between them.
Because the speaker is talking to more than one person.
The grammar here is called the imperative mood. It's what you need when you're telling someone to do something. In English the imperative is pretty easy - there's only one form of it and it's the same as the infinitive form of the verb. In German it's slightly more complicated.
You already know that German has three ways of saying "you" - du for talking to one person casually, ihr for talking to multiple people casually, and Sie for talking formally. This corresponds to three imperative forms, depending on who you're talking to.
Using the example of anrufen, which is a separable verb, we have:
Ruf ihn morgen an - speaking to one person casually
Ruft ihn morgen an - speaking to multiple people casually
Rufen Sie ihn morgen an - speaking formally
Some things to note:
The formal form is the easiest to remember, because it's the same as the infinitive. But we also have to include the pronoun Sie in this situation and not in the others.
The form for speaking to multiple people usually has -t at the end, kind of like how verbs conjugated for ihr often do.
But yes, these are unique forms of the verb that need to be memorised, along with the conjugations and participles.