Is their a way to extract Twitter data as per my requirement i.e. only a small part of data related to my project and store it in my laptop for analysis. If Yes can You please suggest me a way to do it .
You can use twitteR library in R to pull twitter data based on your requirement or to say that #subject name.
You can do so in R as well as in Python. You can extract data related to tweets through API after creating an app.
For R, you can look at this article:
For Python, you can look at this iPython notebook: