Demonstrate how the subjunctive can be used in French?

The Subjunctive in French is often described as a mood rather than a tense. There are many situations in which the subjunctive would be used in French where we would not consider it to be the correct way to translate a sentence from English to French.
It is very difficult to describe the French subjunctive without putting it into context so I propose we work through an example.
The most important way in which the subjunctive is used would be translating sentences such as the following into French: "I want you to do the washing up". If we were to directly translate this sentence into French, it would end up being "Je veux toi faire la vaiselle" which makes absolutely no sense at all to any French person. This is when the subjunctive would be used.
We apply the following formula in order to make this sentence work. We conjugate vouloir in the present tense as normal so we end up with "Je veux", we then add "que" leaving "Je veux que", we then look at the "you" in the english sentence so we add "tu" leaving "Je veux que tu". After this, we now need the subjunctive, so we conjugate faire la vaiselle into the subjunctive tense, making sure it agrees with the pronoun tu, leaving "fasses la vaiselle". Adding this all together we see that "I want you to do the washing up" translates as "Je veux que tu fasses la vaiselle".