When do I use the French subjunctive?

The subjunctive mood (NB: NOT tense) in French is primarily a means of expressing uncertainty - something that you wish, hope or fear, rather than something that is definitely the case. This textbook-style description, though, is a little simplistic and not always useful. For instance, we said that the subjunctive can express hope, but if you want to say 'I hope that you are well', you use the indicative; not the subjunctive (J'espère que tu vas bien). The subjunctive is also often used after 'que' - but, again, this is not the most helpful explanation, tempting students into the trap of peppering every text with unnecessary subjunctives. 

I think the most useful way of looking at the uncertainty element is to take degrees of likelihood into consideration. You might notice espérer, in a likely NEGATIVE statement, taking the subjunctive: e.g., 'Je n'espère pas que Fabrice vienne.' This is a little subjective but, using our best judgement, we might presume that there is little chance of Fabrice coming to the event in question - he clearly hasn't sent an RSVP saying he'll attend. The difference between this phrase and the 'J'espère que tu vas bien above', is that the latter is likely a POSITIVE statement. Again, this is subjective - there are times you tell somebody you hope they're well when, in actual fact, they proceed to tell you that their day has been horrible. Most of the time, though, 'I hope you're well' is a simple conversation starter to which everyone gives a vaguely positive response. A useful breakdown could be to consider that if something is 100% likely to be the case, the phrase will take the indicative; 70% likely, still the indicative; but less than 70% likely and you take the subjunctive. It's not a perfect method, but it tends to work most of the time. 

As for using the subjunctive after 'que', the only way to know whether you should or you shouldn't is to learn by heart the list of conjuctions (il faut QUE, il vaut mieux QUE etc) that apply. It takes time and practice, but the more you use those conjunctions the more natural it will start to feel.