Juan siempre le da una manzana a la maestra.
Juan always gives an apple to the teacher.

What is the direct object? - una manzana

Who received the apple? - la maestra

What is the indirect object? - le, la maestra

"la maestra" is the indirect object because it tells us who the apple was given to.

Another example:

Ella me dio el reporte.
She gave me the report.

What is the direct object? - el reporte

Who received the report? - yo "me"

What is the indirect object? - me

"me" is the indirect object because it tells us who the report was given to.

Long explanation:
An indirect object indicates the aim of the subject's action. It represents the person or thing that is receiving the action. Most of the time it is accompanied by a preposition and it answers the questions to whom?, or for whom?
It can be replaced by the pronouns: me, te, se, le, les, nos.

singular                         plural
me    to me                      nos    to us
te    to you                     os     to you (familiar, Spain)
le    to him, her, you, it       les    to them, you 

se to himself/herself/yourself/themselves/yourselves

A prepositional phrase may be added for clarification or emphasis:

• ¿Me hablas?
  Are you talking to me?

• ¿Me hablas a mí?
  Are you talking to me? [a mí added for emphasis]

• Le dije la verdad.
  I told him/her/you the truth.

• Le dije la verdad a ella.
  I told her the truth. [A ella added for clarification since le can mean “him” or “you”.]

Since these are with-verb pronouns they cannot be used if the verb is only implied. The preposition a plus prepositional object pronouns are used in such cases:

• ¿A quién hablas? ¿A mí?.
  Who are talking to? (To) Me?

• Sí, a tí.
  Yes, (to) you.

Indirect object pronouns are normally used even when a noun is expressed as the indirect object:

• Veo a Carlos.
  I see Carlos. [Carlos is the direct object; lo is not added]

• Le mando un regalo a Carlos.
  I'm sending a present to Carlos.

• Les escribo a todos.
  I write (to) everyone.