Learn Russian: Учитель объясняет мне урок.

мне is the Dative case form of the pronoun я (I). In Russian, the Dative case is used for the indirect object, which is usually the person receiving the action. Since the teacher is explaining the lesson to me, we use мне.

Is the word урок in the Nominative case here? It looks like the dictionary form.

It looks identical, but it is actually in the Accusative case because it is the direct object (the thing being explained). For masculine inanimate nouns like урок, the Accusative form is exactly the same as the Nominative dictionary form.

Can I change the word order and say: Учитель объясняет урок мне?

Yes, you can! Russian word order is very flexible. Putting the indirect object (мне) before the direct object (урок) is just the most common, neutral way to build this sentence. Moving мне to the very end would place a slight emphasis on it, meaning the teacher is explaining it to me specifically.

What is the purpose of the letter ъ in the word объясняет?

That is the hard sign. It doesn't have its own sound, but it stops the consonant б from blending into the soft vowel я. It creates a tiny pause, separating the sounds so they are pronounced distinctly.