Learn Russian: Можешь одолжить мне словарь на урок? - Can you lend me a dictionary for the lesson?

This sentence uses на + Accusative case to show intended duration. While для means "for" in the sense of purpose or benefit, на with the Accusative is specifically used when you take, give, or borrow something for a certain period of time (in this case, for the length of the lesson).

Does the verb одолжить mean "to lend" or "to borrow"?

It can actually mean both! The context and the grammar tell you which one it is. Because we have the Dative pronoun мне (to me), it translates to "lend to me". If you were the subject and just said я хочу одолжить словарь, it would mean "I want to borrow a dictionary".

Why is the perfective verb одолжить used here rather than the imperfective одалживать?

You are asking for a one-time, specific favor that focuses on a successful result (actually getting the dictionary). For single, completed actions like this, Russian uses the perfective aspect.

Why isn't there a ты (you) at the beginning of the sentence?

In spoken Russian, it is very common to drop the subject pronoun if context makes it obvious. The -ешь ending on можешь already makes it perfectly clear that you are talking to "you" (informal singular).