The infinitive of a verb is its "base" form (listed in dictionaries). They are formed by adding the particle to before the verb. Infinitives introduced by to function as noun phrasesadjective phrase, or adverb phrase.

Infinitive = To + Verb

Infinitives in the following sentences are bold. 

To err is human, to forgive divine.
Come to learn, go to serve.
To walk in the morning is good for health.
She wants to be a doctor. 

Types of Infinitives

There are two types of infinitives:

(i) Full Infinitive or To Infinitive

(ii) Bare Infinitive

(i) Full Infinitive

To make a Full infinitive, we add an Infinitive Marker 'to' before the base form of the verb. So, many people like to call it 'To Infinitive'. 

He comes to read.

(ii) Bare Infinitive

Bare Infinitives are infinitives without to.

I told her to sing. (Full Infinitive)
I heard her sing. (Bare Infinitive)

Use of Full Infinitives

You have already known that infinitives are 'to + verbs' that function as nounsadjectives, or adverbs. Here are some examples. 

Infinitives as Nouns: 

Infinitives as nouns does the functions of subjects, objects, or complements. Here are some examples.

SubjectTo read is my gobby.
Object: I love to read.
Complement: My hobby is to read.

Infinitives as Adjective: 

Infinitives as adjectives modify nouns. Look at the following examples.

I don’t have time to read.
The reason to read is to get pleasure.

Infinitives as Adverb: 

Infinitives as adverbs modify verbs, adjectives, or other adverbs. Look at the following examples.

He comes to play with us.
We read to learn. 

Use of Bare Infinitive

You cannot use bare infinitive anywhere you like because there are some rules to use them. Here are they: 

We use bare infinitives after certain verbs like bid, let, make, see, hear, need, dare, etc. If you are learning grammar for exams, you need to commit them to memory.

Let us get up from bed and go out for a walk.
Teachers help us learn.
You need not come with me. 
We make her dance.

The bare infinitive is also used after the Modal Verbs willwouldshallshouldmaymightcancould and must.

You will sing and I shall dance.
None can deny the truth.
One should do one's duty.
You may go now.

The infinitive without to is also used after had betterwould rathersooner than and rather than.

I would rather be a sparrow than a snail. 
You had better go home.