(a)wake and (a)waken

1) forms
The verbs awake and wake are irregular in British English, but can be regular in American English.
(a)wake - (a)woke - (a)woken GB/US English
(a)wake - (a)waked - (a)waked > US English: (a)waken is regular.
(a)waken - (a)wakened - (a)wakened

2) use
Wake is the most common of these four verbs. It can mean 'stop sleeping' or 'make (somebody else) stop sleeping'. It is often followed by up, especially when it means stop sleeping.
I woke up three times in the night.
Could you wake me (up) at half past six?

Waken is a more literary alternative to wake (up):
The princess did not waken for a hundred years.
Then the prince wakened her with a kiss.

Awake and awaken are also rather literary words. They can be used to mean 'wake (up)', but are more often used figuratively, to talk not about waking from sleep, but about the waking of emotions, understanding etc.

I slowly awoke to the danger that threatened me.
At first I paid little attention, but slowly my interest awoke.
The smell of her perfume awakened the gipsy's desire.

3) awake and asleep (adjectives)
Note that in informal British English the adjective awake is more common (in predicative position) than the verb form waking; and asleep is more common than sleeping.
Is the baby awake yet?
You were asleep at ten o'clock.