I'm confused about the use of the conditional tense in German, could you explain how it's formed and its uses?

The first important thing to remember about forming the conditional tense in German is that both parts of the sentence must be in the conditional tense. Take for example the sentence 'If I woke up earlier today, I would not have missed the bus.' Firstly we construct the initial clause: 'If I [had woken] up earlier today'. 'If' becomes 'Wenn' which sends the verb 'to wake up' (aufwachen) to the end of the clause. 'Wenn ich heute früher...' 'had woken up' we take the past perfect form of aufwachen which is 'aufgewacht'. Now the auxiliary verb must be put into the conditional tense. For the verb 'aufwachen' we use the auxiliary 'sein'. 'Sein' in the conditional tense becomes: ich wäre, du wärst, er/sie/es wäre, wir wären, ihr wär(e)t, Sie wären, sie wären. So 'had woken up' becomes 'wäre aufgewacht'. Adding this conditional element to our first clause and applying the rules of word order we get: "Wenn ich früher aufgewacht wäre,..." We do the same thing for the second clause 'I would not have missed the bus'. 'I would not have missed' becomes 'Ich hätte nicht verpasst' adding in 'the bus' becomes 'Ich hätte nicht den Bus verpasst'. Attaching the two clauses together and applying word order rules we get: 'Wenn ich heute früher aufgewacht wäre, hätte ich nicht den Bus verpasst'.