"yo te estoy dando a ti una caja"
In this phrase, which I believe says "I am giving a box to you", what does "ti" represent?
4 Answers
It's redundant, for emphasis I would guess. "A ti" is "to you".
a ti = to you
te= indirect object pronoun for "a ti" 
The a ti is unecessary here because you already have the "te"; it just emphasizes that the box is being given to you . The "te" is required however (see the reference article on indirect object pronouns for more information)
http://www.spanishdict.com/answers/100017/indirect-object-pronouns
it is to specify that the box is being given to you not to someone else, possibly to stress that you are not to give it to anyone else, or that you had asked for it specifically... In english it might sound like "I am giving the box to you alone" (no body else is going to get this box from me)
It just emphasizes that the box is being given to you probably excluding others. In English we have to express that emphasis with tone of voice. It also allows Spanish to have a written mechanism to convey it. (where we would have to bold or italicize the word, etc.)
I am giving you the box.
