An auxiliary verb is one which helps other verbs form tenses other than the simple present and simple past.

The following are the auxiliary verbs in English:

Be and its forms: is, am, are, was and were
Do and its forms: does, did
Have and its forms: has, had
Shall, should; will, would
Can, could; may, might
Must; ought; dare; need

Read the sentences given below.

Alice is reading a novel.
I am cooking dinner.
They were playing.
He has done wrong.
We have worked really hard.
They did not come in time.
How are you?
What did she say?

You will have noticed that in the sentences given above, the auxiliaries be, do and have merely express statements of fact. These three auxiliaries and their various forms are called primary auxiliaries.

Now read the following sentences:

He will come.
He may come.
She must come.
I should go.
You ought to go.
You need not go.

These sentences are not statements of fact. They, in fact, express actions or events that exist as conceptions of the mind – probabilities, possibilities, obligations, wishes, expectations etc. Several factors may prevent these actions or events from being fulfilled. Auxiliaries which help to express such  conceptions of the mind are called modal auxiliaries.

Modal auxiliaries help to express the subjunctive and imperative moods. Note that sometimes modal auxiliaries are used to express simple futurity. Similarly, it is also possible for primary auxiliaries to be used with modal force.

Uses of the Primary Auxiliaries

The primary auxiliaries (be, do and have) are used both as principal verbs and as auxiliaries. They are auxiliaries when they help other verbs to form their moods and tenses.

Read the sentences given below:

Alice is my sister.
She is knitting.

In the first sentence is is a principal verb as it says something about the subject without the help of any other verb. In the second sentence is is an auxiliary verb helping to form the present continuous tense of knit. More examples are given below:

She has two children.
They have returned home.

In the first sentence has is the principal verb. There are no other verbs in that sentence. In the second sentence have is an auxiliary verb helping to form the present perfect tense of return.

She did her work well.
When did he return?

In the sentence She did her work well, did is the principal verb whereas in the sentence When did he return, did is an auxiliary verb helping to ask a question in the simple past tense.