Learn English – Conditional sentences not starting with “if”

What is called the "subjunctive" or the "present subjunctive" is in fact just a simple untensed variant of normally tensed that-complement clauses.

Unlike real subjunctive systems in languages like German or Spanish, this construction appears
- only in subordinate complement clauses,
  never in main clauses, or other kinds of subordinate clauses;
- only in one variety of subordinate clause (that-complements);
- only with one variety of predicate (impositives).
It's what you call extremely limited in distribution.

Complement clauses are governed by their predicates. Which predicate you use controls what kind(s) of complement can be used with it, or whether it can have a complement at all. With impositive predicates (see lists below), that-complements must have a subject noun phrase (i.e, the subject NP can't be deleted), but in these that-clauses the verb form is infinitive, not present or past; i.e, the verb does not agree with its subject NP.

Since this construction is governed by the predicates it can occur with, and the predicates are all impositive -- i.e, they are predicates indicating that the speaker is imposing their will on the addressee -- the construction is associated with that concept, and traditionally "subjunctive" verbs were sposta "express" this "mandative" notion. Actually, of course, that's expressed by the predicate, and only emphasized by the use of this construction with it.

There are four patterns, with four different kinds of complement-taking impositive predicates:

1) Transitive impositive communication verbs:
insist, suggest, demand, prefer, propose, suggest, recommend, demand, ask, mandate, prefer, request, ask, desire, advise, urge, specify, instruct, order, demand, insist, require, rule, necessitate, suffice, advocate, vote, would rather, and move (in the parliamentary sense).

Pattern: NPVolit Verb that [NP + Infinitive VP]
They asked that he remove his shoes.
They prefer that she arrive ten minutes early.

2) Transitive impositive emotive predicate adjective: adamant

Pattern: NPVolit be Adj that [NP + Inf VP]
They were adamant that he remove his shoes/that she arrive ten minutes early.

3) Intransitive impositive predicate adjectives (normally with Extraposition):
necessary, desirable, imperative, important, necessary, preferable, optional, permissible, acceptable, okay, all right, satisfactory, desirable, advisable, sufficient, necessary, mandatory, urgent, vital, crucial, essential, fitting, right, appropriate, better, expedient, and legitimate

Pattern: That [NP + Inf VP] be Adj = Extraposition => It be Adj that [NP + Inf VP]
That he remove his shoes is desirable = Ext => It is desirable that he remove his shoes.
That she leave early is preferred = Ext => It is preferred that she leave early

4) Picture nouns derived from impositive predicates:
recommendation, necessity, insistence, proposal, preference, request, desire, advice, suggestion, option, alternative, recommendation, demand, requirement, necessity, imperative, condition, mandate, specification, rule, ruling, edict, instruction, principle, prerequisite, order, qualification, ultimatum, vote, and motion (in the parliamentary sense).

Pattern: [PictureN that [NP + Inf VP]] (not a clause, but an NP with a complement clause)
I heard about [the request that he remove his shoes].
[The request that she arrive ten minutes early] is highly improper.

There are other predicates, but if one learns these lists, one comes to recognize the semantic characteristics of impositive predicates, and thus to decide whether to use this construction with them, or others, or not. That's it for the "present subjunctive".

The "past subjunctive" -- which I will not deal with here -- is a different set of constructions altogether, completely distinct from the "present subjunctive" constructions, expressing nothing like their sense, and behaving in completely different ways.

The reason why they're both called "subjunctive" is because Latin had a subjunctive mood that was used in many ways, and these constructions reminded classically-educated grammarians of certain uses of the Latin subjunctive mood.

But in English they're just a couple more mildly idiomatic constructions, out of thousands. That's why obsessing about the English "subjunctive mood" is a waste of time and effort, and always ends up in confusion.

Just as Fumblefingers comments, as long as there is a conditional clause (ie a subject-verb combination beginning with if, unless, etc) in a sentence, whether it follows the main clause or precedes it, it is always a conditional sentence.