Hallo PaulS108 and sfpugh,
The three verbs hören, anhören and zuhören all mean to listen but cannot be used interchangeably, so it depends on the context which word is more appropriate to use. I can totally understand that this can be confusing for a non-native speaker.
Hören means to hear or to listen in general, for example:
“Ich höre gerne Musik.“ - “I like listening to music.“
“Ich kann dich hören.“ - “I can hear you.“
Anhören or (sich etwas) anhören means to listen (to something or someone). It is more active than hören but not as active as zuhören. You might actively listen to something or someone but there is no response or interaction expected from you, for example:
“Ich hörte mir das Konzert an und ging nach Hause.“ - “I listened to the concert and went home.“
“Ich höre mir die neue CD an während ich koche.“ -“ I’m listening to the new CD while I’m cooking.“
Zuhören or (jemandem) zuhören means to listen (to someone). It’s about being attentive, concentrate and trying to understand what the other person has to say. You listen carefully and then reflect on what the other person has said, for example:
“Ich höre dem Professor zu und mache Notizen.“ - “I’m listening to the professor and taking notes.“
“Sie ist ein guter Zuhörer.“ - “She is a good listener.“
There is usually some sort of outcome such as learning something new, asking a question or responding to the other person in some way.
I hope this helps!