Learn Russian: Сколько тебе лет?

In Russian, you don't say "I am X years old." Instead, age is thought of as something that happens to a person. Because of this, you use the Dative case (which points out "to whom" something happens). тебе means "to you," so the sentence literally translates as "How many years to you?"

The word for "year" is год, so why does this sentence use лет?

The question word сколько (how many) requires the noun that follows it to be in a plural quantity form (the Genitive plural). For the word год, this plural form completely changes to лет (which historically meant "summers"). You will also use лет when stating ages 5 and up.

Where is the verb "are" in this sentence?

In Russian, the verb "to be" is almost always left out in the present tense. While in English we ask "How old are you?", Russian simply asks "How many to you years?" without needing a verb at all.

Does the word order have to be exactly Сколько тебе лет?

Russian word order is very flexible. You will also frequently hear Сколько лет тебе?, which keeps "how many" and "years" next to each other. Both are perfectly correct, but Сколько тебе лет? is the most common conversational flow.