The subjunctive isn't a tense, it's a mood, used for things that aren't stated as plain facts: wishes, obligations, feelings, doubts. It almost always shows up after que, with a different subject in each half of the sentence.
1. Formation: the ils-stem + subjunctive endings
Take the present-tense ils form, drop -ent, and add -e, -es, -e, -ions, -iez, -ent.
| parler → ils parlent → stem parl- | que je parle, que tu parles, qu'il parle... |
| que nous parlions, que vous parliez | that we speak, that you speak |
2. Irregular subjunctives
| être | que je sois, que tu sois, qu'il soit, que nous soyons... |
| avoir | que j'aie, que tu aies, qu'il ait, que nous ayons... |
| aller | que j'aille, que nous allions... |
| faire | que je fasse, que nous fassions... |
3. When to use it: wish, necessity, emotion, doubt
| Il faut que tu partes. | You have to leave. (necessity) |
| Je veux que tu viennes. | I want you to come. (wish) |
| Elle est contente que nous soyons là. | She's happy we're here. (emotion) |
| Je doute qu'il ait raison. | I doubt he's right. (doubt) |
4. Watch out: same subject → infinitive, not subjunctive
Common mistake: the subjunctive after que requires two different subjects. When the subject is the same in both parts, French drops que and uses an infinitive instead: Je veux partir. (I want to leave, not "je veux que je parte") vs. Je veux que tu partes. (I want you to leave.)
5. Watch out: espérer que takes the indicative
Despite expressing a wish, espérer que (to hope that) is a well-known exception, it's followed by the regular indicative, not the subjunctive.
| J'espère que tu vas bien. | I hope you're doing well. (indicative, not "que tu ailles") |