Bill Franke's answer is admirable, and this is not offered as an alternative; but there's a lot of historical and technical baggage behind the question of the English “subjunctive”, so a little clarification may be in order.
At bottom it's a terminological dispute.
Subjunctive in Latin (and Greek) grammar was the traditional name for distinct inflections of the verb used in specific syntactic and semantic contexts. Since in most of these contexts the clause which the verb headed was subordinate, explicitly or implicitly, to a main clause, these forms were given a name which indicated that they “subjoined” the clause; the distinction was categorized as one of mode or ‘mood’, opposed to other moods expressed with other forms: indicative, imperative, optative, infinitive are the terms employed by Dionysius Thrax in the 2nd century BCE.
Many European languages maintain a distinctive set of forms employed similarly to Latin which it is not too far-fetched to call subjunctive. It is quite otherwise with English. In Present-Day English only one verb, be has as many as eight forms; most have only four or five; a handful have only one or two. In these circumstances the notions of subjunctive and of mood itself take on very different significance. Early (17th- and 18th-century) students were puzzled how to apply the Latin terms to English: some recognized mood as a category realized with auxiliary verbs, others denied the existence of mood altogether:
Now in English, there are no Moods, because the Verb has no Diversity of Endings, to express its Manners of signifying; but does all that by the Aid of Auxiliary or Helping Verbs which in the Latin, and some other Languages, is done by the Diversity of Terminations or Endings. – Greenwood, An Essay towards a Practical English Grammar (1711)
(Here is a fascinating study of 18th-century treatments of the subjunctive.)
Nonetheless, by the 19th century it had become usual and convenient to acknowledge a subjunctive mood in English, analogous to Latin, and to understand the specific forms elicited in subjunctive contexts as subjunctive forms. (I suspect this is because until very recently formal English grammar was taught primarily within a context of Latin literacy, and opinion gradually hardened around pedagogically useful concepts which aligned Latin and English usage.)
Over the past fifty years, however,formal grammarians have turned their backs on this approach and endeavoured to develop models and terminology specifically suited to the description of English. Consequently, there are today a number of notions of just what the English subjunctive is. For those like me and Bill Franke, who were taught in the Old Style, subjunctive is in the first instance a name for the verb forms. For some formal linguists, subjunctive is a name for the modal context which elicit those forms: these will tell you that “Although English clauses may be subjunctive, English verbs are not”. And other linguists argue that the term itself is useless; other, happier terms account quite parsimoniously for most grammatical phenomena, and subjunctive survives in contemporary linguistics mostly as a name for the use of be in contexts such as that which the current question concerns—and even there it is often deprecated.