What is the subjunctive and what is it used for?

The subjunctive is a grammatical mood meaning that it signals the intention of the speaker (or writer) in what they say. Other moods include the indicative (statements of fact), the interrogative (direct questions) and the imperative (direct commands). The subjunctive is used to express things that are not necessarily real, being hypothetical or unconfirmed. It does exist in English but is not especially obvious and most monolingual English speakers wouldn't even think of it. For example:
John suggested that Jack apologise
This is subjunctive because it us unknown to us whether or not Jack actually does apologise and the prospect of Jack apologising remains a suggestion.
The subjunctive is used for a variety of reasons in Latin including :The expression of wishes and exhortations (Let's feed the dog)Hypothetical statements that aren't considered to be fact by the speaker (The dog might be hungry)Indirect questions and commands - when the speaker recounts questions asked or commands given (He asked me to feed the dog)Purpose clauses which express the purpose for which something is done (I went home to feed the dog)Result clauses which express result of something being done (The dog was so hungry that he ate very quickly)Fearing clauses which express fear that something may or may not happen (I'm worried that the dog hasn't been fed)Closed conditional clauses which create hypothetical statements based on a condition which isn't fulfilled (This wouldn't have happened if you'd fed the dog)
All these function in different ways in Latin. The examples are simply intended to introduce you to the various uses and what they mean before learning them individually in Latin itself.