1. 1. The Synthetical Subjunctive 2. The Analytical Subjunctive
The Subjunctive
Unlike the Indicative Mood,  which expresses a real state or event, the Subjunctive Mood expresses an unreal, unaccomplished state or event.
There are two kinds of subjunctive:
1. The Synthetical Subjunctive
2. The Analytical Subjunctive

2. 1. The Synthetical Subjunctive
The Synthetical Subjunctive has the following forms:
a. The Present Subjunctive
b. The Past Subjunctive
c. The Past Perfect Subjunctive

3. a. The Present Subjunctive
This subjunctive is formally identical with the short infinitive.
It expresses events or states which are assumed possible;
It can convey wishes, orders, requests, conditions, concession etc.
It occurs in independent sentences, main clauses and subordinate clauses

4. The Present Subjunctive
Use:
in independent sentences (exclamatory sentences) or main clauses, to express a wish or hope (Formulaic Subjunctive):
Long live the Queen!
Heaven help us!
Be that as it may!
So be it!
Come what may, we’ll be together!
Bless you!
Suffice it to say that my stay here was a pleasurable experience.
Damn you!
“If this be error and upon me proved/I never writ nor no man ever loved.”
(Sonnet 116, William Shakespeare)

5. The Present Subjunctive
2. in direct object clauses, after verbs such as demand, command, order, request, suggest, urge etc., to express obligation, necessity, requirement:
The manager requested that all employees be present at the meeting.
The doctor suggested that grandma stay in hospital for full examination. 
3. in subject clauses, after impersonal constructions such as it is/was advisable/important/necessary/essential etc., to express obligation, necessity, requirement:
It is essential that she understand the situation.
It is important that he know the truth. 
4. in conditional sentences, to express condition or assumption:
If need be, we’ll support you.
Supposing she come, what will happen?

6. b. The Past Subjunctive
The past subjunctive is formally identical with the past simple .
The verb to be has the form were throughout. In the informal style, was is more usual, but this may lead to confusion:
-the past subjunctive refers to unreal situations or events;
-the indicative mood refers to real events.
He behaves as if he were the boss. (he isn’t the boss)
He behaves as if he he‘s having trouble with the new machine. (he is having trouble)
The past subjunctive, also known as unreal past,  expresses an unreal, improbable event or state occurring at present, running counter to reality.

7. The Past Subjunctive
Use:
The past subjunctive appears in the following kinds of subordinate clauses:
in conditional sentences:
If I were you, I would be more careful.
They would be more helpful if you treated them with respect. 
2. in direct object clauses, after the verb wish and phrases like would rather and would sooner:
I wish he were here now.
Our parents wish you were telling the truth.
I would rather you left now.
would rather/sooner can also be followed by short infinitive if the subject of would rather/sooner is identical with the subject of the second verb:
I’d rather leave now.
He’d sooner wait for you.

8. The Past Subjunctive
3. in subject clauses, after it’s (high) time:
It’s time we were leaving.
It’s high time the children went to bed.
It’s (high) time can also be followed by long infinitive or for + object + long infinitive:
It’s time to leave.
It’s time for the children to go to bed.
There is a slight difference in meaning between the two constructions:
It’s time + past subjunctive states that the correct time has passed, it is a little late.
It’s time + infinitive states that the correct time has arrived.

9. The Past Subjunctive
4. in adverbial clauses of comparison, after as if, as though:
You look as if you were ill. (but you are not ill)
They were speaking to as though they knew me. (but they did not know me)
5. in adverbial clauses of concession, after even though, although, though:
Even though she were punished, she wouldn't give me away.

10. c. The Past Perfect Subjunctive
The Past Perfect Subjunctive is formally identical with past perfect tense. It expresses an unreal action running counter to a past event or state.
Use:
The past perfect subjunctive is used in subordinate clauses. The action expressed by the past perfect subjunctive can be either simultaneous to the action in the main clause or prior to it.
in conditional sentences:
If he had driven more carefully, he wouldn't have had the accident. (but he didn't drive carefully) 
2. in direct object clauses, after the verb wish:
I wish I had accepted their proposal. (but I didn't) 
3. in adverbial clauses of comparison, after as if, as though:
I fell as if i had been hit by a truck. (it didn't hit me) 

11. 2. The Analytical Subjunctive
The Analytical Subjunctive is made up of modal auxiliaries followed by the  indefinite or perfect infinitive of the lexical verbs.
The indefinite infinitive expresses simultaneity or posteriority to the action in the main clause.
The perfect infinitive expresses anteriority to the action in the main clause.

12. a. Should + infinitive
a. Should + infinitive is used both in main and subordinate clauses:
1. in main clauses expressing emotional attitudes, after why and how:
Why should we fight over this? 
2. in main clauses, to express a present conditional:
I/We should like to talk to him.I/We should have liked to talk to him. 
3. in subordinate clauses of condition to denote an unlikely event:
If you should meet Henry, tell him to phone me.
(If you happen to meet Henry ... )

13. Should + infinitive
4. in subject clauses, after
it is /was + adjective:
it is/was fortunate/important/ impossible/ natural/ necessary/ strange/ surprising/ unusual
it is /was + noun:
it is/was a pity/shame/surprise/wonder
It is surprising that he should be such a nuisance.
It's a pity that they should leave so early. 
5. in direct object clauses, after verbs such as agree, demand, command, insist, offer, order, propose, request, suggest, urge etc., to express obligation, necessity, requirement:
We requested that the reports should be presented at our next meeting.

14. Should + infinitive
6. in affirmative adverbial clauses of purpose introduced by so that, in order that, that and negative adverbial clauses of purpose introduced by lest , for fear that:
They left me a note in order that I should know what they had done.
I took a taxi lest I should miss the train. 
7. in adverbial clauses of concession introduced by although, though, whatever, no matter:
No matter how hard it should be, we'll perform our task. 
8. in adverbial clauses of time:
Keep trying until you should make sure you do everything well. 

15. Should + infinitive
9. in prepositional object clauses, after adjectives such as to be glad, to be anxious, to be pleased, to be sorry etc.:
I am sorry that you should feel this way. 
10. in relative clauses, after nouns such as hint, idea, reason, supposition, thought:
There is no reason why you should behave like this.

16. b. May/might + infinitive
May/might + infinitive occur in main and subordinate clauses.
May or might + infinitive can be used when the verb in the in the main clause is in a present or future tense.
Only might + infinitive is used when tha verb in the main clause is in a past tense.
Use
in main clauses expressing a wish:
May you live long!
May all your dreams come true!
Oh, that she might be here before dawn.

17. May/might + infinitive
2. in subject clauses after it is/was possible/probable/likely:
It is possible that they may visit us tomorrow.
It was likely that they might have been there before. 
3. in prepositional object clauses after be afraid:
I'm afraid they may/might be late.
He was afraid we might reveal his secret. 
4. in adverbial clauses of purpose introduced by that, so that, in order that:
I'll give you some money so that you may buy yourself a new suit.
They asked for more information in order that they might fill in the applications. 

18. May/might + infinitive
5. in adverbial clauses of concession introduced by although, though, whatever, however, no matter:
However hard it may be, we must succeed.
No matter how mentally exhausted he might have been, he still had to finish the experiment. 
6. in direct object clauses, after wish to express an unfulfilled action:
I wish they might have understood me.

19. c. Would + infinitive 
Would + infinitive occurs both in main and subordinate clauses.
Use:
1. in conditional sentences to express a present or a past conditional:
He would come to your party if you invited him.
I would have understood you if you had explained everything to me. 
2. in direct object clauses, after the verb wish, to complain about an annoying habit or to refer to something the speaker would like to happen:
I wish my children would stop making so much noise.
Jerry wishes Ann would talk to him more often. 

20. would + infinitive is used when the main verb is in a past tense.
3. will/would + infinitive may be used in adverbial clauses of purpose introduced by so that:
Mrs. White left the man waiting at the door so that she would  announce her husband about the visit.
He volunteers at the homeless shelter so that he will help those most in need
will + infinitive is used when the main verb is in a present, present perfect or future tense.
would + infinitive is used when the main verb is in a past tense.

21. d. Could + infinitive Could + infinitive occur in subordinate clauses.
1. can/could + infinitive may be used in adverbial clauses of purpose, after so that, as an informal alternative to may/might + infinitive; could + infinitive implies that the action is viewed as real:
I lent them a good dictionary so that they could translate the article in French.
can + infinitive is used when the main verb is in a present, present perfect or future tense.
could + infinitive is used when the main verb is in a past tense.
2. could + infinitive is used in direct object clauses, after wish, to refer to something that is difficult or impossible to accomplish:
We wish they could get in touch with us. 

22. REMEMBER!
The Synthetical Subjunctive is mainly used in legal documents and journalism. It is more frequently used in American English than in British English.
The Analytical Subjunctive is more used in contemporary British English.