Hasn't, doesn't.

[essentuki]

Sorry but I do not know what the title should be called?

Have I written the 2 words correctly?

Hasn't equates to 'has not', doesn't equates to 'does not'.

There are many other examples of replacing a missing letter with an apostrophe, what is the correct term for such usage?

Is its use correct?

Jeff Gordon.

[Maeve]

Jeff,
Here's a DWT post that addresses the use of the apostrophe in forming contractions:

http://www.dailywritingtips.com/contractions/

Maeve