The topic of the Subjunctive Mood is generally presented as part of the most advanced-level Spanish courses. The typical introduction would say that the Subjunctive Mood in Spanish is used to express: “wish, conjecture, emotion, uncertainty, subjectivity, requests, doubt.” The subject is considered to be complex and is usually accompanied by a correspondingly formidable introduction, such as the one below:
Another way of expressing the concept is that the indicative expresses reality or what is believed to be reality. But the subjunctive is used for different purposes: It expresses facts that are contrary to reality. It expresses doubt that something is or will be a fact. It expresses how a person feels about a possible action or state of being. It expresses a wish, intent or command for a possible action or state of being. 
In many ways, the above quote is correct, and, since subjunctive in English is not as common , understanding the subtleties of how it’s used in Spanish does take preparation and is best left for advanced sections of the course; however, there is one type of very simple English sentence that is always expressed by subjunctive in Spanish. Here are a couple of examples:
Sentences of this kind are usually tucked away with other subjunctive topics under obscure headings like “subject change in dependent clauses,” or “request/instruction subjunctive indicator,” but they are a very basic type of sentence you need to know in order to be even moderately functional in Spanish.
Fortunately, unlike other uses of the subjunctive, this one is very straightforward — an English sentence of the type “Juan wants someone to do something” or “I asked someone to do something” will always be translated into Spanish using subjunctive according to two simple rules discussed below.
I believe sentences of this kind are the easiest and most practical way to introduce the subjunctive to intermediate students — they are easy to learn, and, more importantly, they are immediately useable in everyday situations. The big advantage is that their usefulness will give the students a strong incentive to learn the subjunctive form quickly, and since they will find it easy to employ this aspect of the subjunctive in their daily lives, some of the unnecessary mystery surrounding the topic will be dispelled. This way, by the time more complex uses of the the subjunctive are introduced, the students will have had plenty of practice with simple sentences such as, “Hey, could you ask Juan to buy some fruit?” or “Do you want María to send you an email?”
Demystifying the subjunctive: The tense and moody verbs
One of the things that makes learning the subjunctive difficult is that the names of the subjunctive forms (e.g. Present Subjunctive, Past Imperfect Subjunctive) encourage you to think that they are a type of tense. But they are not tenses — they are moods. What does that mean exactly? To answer, let’s take a more detailed look at one of the examples from above.
In this case, the verb “venir” is in the present subjunctive form “vengas” rather than in the regular, indicative form “vienes.” A standard explanation for the use of the subjunctive is that this sentence expresses “wish” or “desire” and hence the verb “venir” must be in this special form. But logically this seems to make little sense — after all, the “I want you to” part already expresses the wish or desire quite clearly. Why do I have to express it a second time?
It helps to move the shoe onto the other foot to fully grasp the idea of the subjunctive. Let’s look at things from the perspective of a Mandarin Chinese speaker who is starting to learn English and consider the following simple sentence:
In Mandarin, the past tense is not used in sentences of this kind, so in a Chinese speaker’s mind, this idea has the form “Yesterday I see a movie.” The two of you could conceivably have the following exchange:
If you think about it logically, the sentence “Yesterday I see a movie” is perfectly complete; it tells that the action has already happened, but yet in English this sentence sounds wrong.
Something similar happens with using subjunctive in Spanish: if you forget to use it, it is still clear that you want the person to come to the party, but it is still a serious grammatical error.
As one more example, let’s consider the following two sentences in English:
The sentence (A) says that Mary believes that John is on time, and furthermore, she is insisting on the truth of that statement. The sentence (B) implies that John is often late and that Mary is demanding that John come on time more often. The reason the meaning changes so drastically is that this is one of the rare cases where the Subjunctive Mood is used in English. By changing from “John is on time” to “John be on time” we change the meaning from stating a fact to desiring a particular outcome. This usage is fairly common in writing, especially to express strong demands or urgency, e.g. “The quiz show organizers demanded that the rules be followed.” (More examples here) The structure of sentence (B) is very similar to the structure that is used in Spanish, except in Spanish it is used all the time.
In summary, it’s important not to be misled by the names of the Subjunctive forms and regard them as just another grammatical tense. With tenses, we express the “when” of the action in question; with the subjunctive mood, we express how we feel about it.
Two Rules for Simple Introduction of the Subjunctive
Returning to brass tacks, let’s look the rules that allow the Subjunctive to be introduced in a very simple way. Basically, in sentences that express requests, the English infinitive is always translated as corresponding subjunctive in Spanish. From this, two rules follow:
Rule I: If we want to say that person A asks/wants/tells person B to do something, we must use the present subjunctive (el presente de subjuntivo) in the second part of the sentence.
Rule II: If want to say that person A asked/wanted/told person B to do something, we must use past imperfect subjunctive (el pretérito imperfecto de subjuntivo) in the second part of the sentence.
Important Note: In English, when we change the request from the present to the past, the second part of the sentence stays the same: “she wants them to do it” –> “she wanted them to do it,” but in Spanish it changes “ella quiere que lo hagan” –> “ella quería que lo hicieran.”
1. When to Use the Subjunctive Mood
(a) To express conditions that are contrary to the fact, e.g. “If I were you” or “If he were here.” In such cases, the verb “to be” takes the form “were” for all subjects whether singular or plural.
(b) In sentences that express insistence, urgency, requests, e.g. “They insist that he stop smoking.” In this case, the subjunctive form is only visible in the third person singular where the normally present “s” is omitted.
More on this: http://www.englishclub.com/grammar/verbs-subjunctive.htm
3. In Spanish, the subject pronoun is usually omitted since the verb form already implies who the subject is. In such cases, I indicate the subject in parentheses for completeness.