How do I form the subjunctive and when is it used?

The subjunctive verb form is what is called a 'mood' - it is not a tense like the passé composée or the imparfait, but it is used to indicate feelings, thoughts, doubts and anything that is subjective in nature, and is formed differently to the indicative (the 'mood' of all the verbs you will have seen previously). The subjunctive in French can be a little tricky to native English speakers as it is not used often in English, especially not in informal or spoken speech.
To form regular subjunctive verbs: 1. Take the third person plural form of the verb (the ils/elles form) in the present, e.g. ils finissent 2. Take off the '-ent' at the end to get the subjunctive stem, e.g. ils finiss- 3. Add the subjunctive endings, which are: Je finisse Tu finisses Il/elle/on finisse Nous finissions Vous finissiez Ils/elles finissent
Most structures that require the subjunctive include the word 'que,' and many are used to express preferences, advice, needs, desires or orders (just remember the acronym PANDO!), e.g. Je préfère que + subjunctive (I prefer that...) Je craigns que/j'ai peur que + subjunctive (I fear that...) Il est important que + subjunctive (It is important that...) Il faut que + subjunctive (It is necessary that...) J'exige que + subjunctive (I demand that...)