Why and how was the apostrophe invented?

Hint: An apostrophe is a symbol (’) used in writing to indicate the omission of a letter or a number. For e.g. in I'm (= I am) or '85 (= 1985). It is also used before or after s to show possession, as in Helen's house or babies' hands.

Complete answer:
The first time when the use of an apostrophe was described, was in 1588. We find this information from a book called "Making a Point" written by David Crystal.
The apostrophe was used for the first time by Pietro Bembo (an Italian Scholar) in his edition of De Aetna (1496). It was introduced into English much later, in the 16th century and is claimed to be done in imitation of French practice.

Some examples of apostrophe are:
We spent Christmas Day with Rahul’s parents.
This book is the fruit of 20 years' research.
Rajesh has a clear-sighted vision of the company's future.
Our neighbor's baby cries morning, noon and night.
I'm sure my views on many things are colored by my parents' divorce.

In the above examples we can see that we have used apostrophes in the underlined places. There is on simple rule of English which is followed while marking apostrophes is that whenever there is already “s” at the ending of the word we put the apostrophe after the s but when there isn’t s at the ending we simply put an apostrophe after the word and then put s. We can see in the examples that Rahul, company, neighbor do not have s whereas years and parents have an s.

Note:
i) From the 16th century, inspired from the French practice, the apostrophe was used whenever a vowel letter was omitted, because of incidental elision ("I'm" for "I am") or because the letter no longer had a sound of its own ("lov'd" for "loved").
ii) English spelling retained many inflections that were not pronounced as syllables, notably verb endings ("-est", "-eth", "-es", "-ed") and the noun ending "-es", which marked either plurals or possessives, also known as genitives.