The present subjunctive is a verb form used in certain constructions to express a "non-assertion". In simple terms, it is generally used in constructions that do not simply "state that something happens or will happen" (assertion). A clue to something being an assertion is that it can generally be contradicted using the formula no, ... doesn't/didn't. For example, in:
(1) the boss said (that) Susan worked hard
we could reply with either no, he didn't! or no, she didn't! to agree with either of the two assertions in the sentence. But in:
(2) the boss demanded that Susan worked hard
only the first of these contradictions, no, he didn't!, makes sense. This indicates that that Susan worked hard is not an assertion in this case. (Another feature of non-assertions in English is that that often cannot be omitted, whereas, for example, in sentence (1) it can.)
In Spanish, clauses that express a non-assertion-- such as that Susan worked hard in (2)-- generally have their verb in the subjunctive. Despite its name, the present subjunctive is generally the form used when this non-assertion is "non-past" rather than specifically "present".
Typical constructions involving the present subjunctive include:
Typical constructions that don't trigger the subjunctive include verbs/expressions of simple 'saying or quoting' such as decir que ..., añadir que ..., es muy claro que ..., su declaración de que ... (his declaration that ...) etc.
A common but misleading perception is that the subjunctive is somehow a "learned" form, only used in "advanced constructions". In fact, subjunctive forms in Spanish are common, everyday forms used naturally in very frequent, ordinary constructions. For example, cuándo with a future time reference, or expressions such as no creo que... (I don't think that ...), are perfectly common in everyday speech (and generally trigger a subjunctive verb).
English equivalents to the subjunctive
English does not have subjunctive verb forms (although Old English did), and instead comparable constructions in English use other formulae such as:
that X might Y
that X should Y
that X would Y
(for) X to Y
X -ing Y
As in sentence (2) above, English does not always have an obviously special construction to express a non-assertion. (Though as we mentioned, one special feature of (2) is that the word that is obligatory.) In cases such as (2), some dialects of English also use a so-called "null modal" construction (demanded that she work hard, in which she is followed directly by an infinitive, as though there is an "empty modal" verb between the two).
Formation of the Spanish subjunctive
Formation of the subjunctive is generally predictable if you know the present indicative (i.e. the "ordinary" present tense):