An auxiliary (or "helping") verb is a verb that doesn't really mean much at all, but helps with the grammar.
For example, take this sentence:
Sally did not pass the exam.
What does "did" mean in that sentence? Well, nothing; but it so happens that in English, if you want to make a negative statement in the past simple tense, you need the extra verb "did", the past tense of "do" -- it's an auxiliary verb.
This is a trickier case:
How do you do?
In this sentence, the first "do" is an auxiliary verb -- it is merely required to form a question in the present simple. The second "do" is the main verb.
You have to be careful; in English, auxiliary verbs also double as main verbs.
The English auxiliary verbs are:
do -- used to form questions and negatives in the simple tenses
have -- used to form the perfect tenses (I have made -- I had made)
be -- used to form the continuous (or progressive) tenses (I am making -- I was making) and the passive (it is made -- it was made)
Remember that these verbs also function as main verbs: "do" is a general-purpose verb (e.g. I did my homework), "have" can mean "possess" (I have a car), "be" can mean "exist" or "behave" and many other meanings besides (I am English). They are only auxiliary verbs when they serve only grammatical purposes.
In addition, the so-called modal verbs also function as auxiliary verbs (and so are sometimes called "modal auxiliaries"). These indicate possibility or obligation, and include will/would, can/could, shall/should, may/might, must and ought to.