What do you call the person who does the verb? For instance, in the sentence
John killed Frank
what is the grammatical term for John?
I don't know if "agent" is the right word?
I take it that you are asking for a technical term used in the formal linguistic study of grammar.
Within that domain, the questions you ask in the title and in the body of your question are really two different questions, and require two different actions. Permit me to rephrase a little.
What do you call the person who ‘does’ the verb [in a sentence]?
What do you call the person who performs [the] action [expressed in the verb]?
It is important to distinguish these two entities, because in some sentences they may be different persons (or animals or institutions or in fact any noun or noun phrase – they need not be persons).
In the sentence “John killed Frank”, for instance, John is both the subject of the verb kill and the Agent of the action the verb expresses. (Frank, by the way, is the direct object of the verb and the Patient of the action it expresses.)
But if you recast the same sentence into the passive voice, “Frank was killed by John”, Frank (which was the direct object in the previous sentence) becomes the subject of the verb—but John is still the Agent of the action.
The noun (person, place or thing) that performs the action directed by the verb is the subject of the sentence. For example:
John is the subject and walks is the verb.
If you mean the noun that the verb is done to, then it's the direct object. For example:
The dog is the direct object.
Previous answers have covered the question in terms of traditional grammar, but another way of looking at language is from the perspective of functional grammar. It’s a complex subject, but this is how functional grammar would analyze your example.
Functional grammar sees a clause as centred on a Process, realised as a verb. Processes can be of several kinds, but the one we perhaps come across most often is the Material Process, that is, something that happens in the real world. In the example, killed is just such a Material Process. The persons and things that initiate a Process and that may be on its receiving end are called Participants. Where the Process is a Material Process, the initiating Participant is known as the Actor, and the Participant being acted upon is known as the Goal. So, in the example, John is the Actor and Frank is the Goal.
As StoneyB has pointed out, making the clause passive doesn’t change the fact that it was John who did the killing. Functional grammar recognises this in that in the passive clause Frank was killed by John, John is still the Actor, and Frank is still the Goal.
The subject, I suppose. You know, the subject performs the action contained in the verb. Example: The cat [subject] chased [verb] the mouse.