Questions & Answers about Я жду тебя у входа в библиотеку.
The verb is ждать (to wait (for)). In the 1st person singular present tense it conjugates as я жду. Many Russian verbs don’t form the 1st person singular by simply adding -ю to the stem; ждать → жду is an irregular-looking but standard form.
Ты is the subject form (you). Here, you is the person being waited for, i.e., the direct object, so Russian uses the accusative case: ты → тебя.
So я жду тебя = I’m waiting for you.
No. Russian ждать normally takes a direct object without a preposition: ждать кого/что (to wait for someone/something). That’s why it’s жду тебя, not жду для тебя or similar.