1) (Being a vegetarion) I am a vegetarion ...
2) Being a Doctor has its ups and Downs (I am Doctor)
3) I apologize for not being able to attend the marriage (I am not able to)
4)  Would you mind being quiet for a moment?
5) My daughther enjoyed being taken at the circus..
6) Thank you for sending me a beautiful greeting card.
In all the above sentences being, a gerund, is used.
1) I get little bit confused, when I have to use word, being, in sr. No. 3,4 and 5 and when not, like in Sr. no. 6. In Sr. No. 6, there is no use of word,being. I am not aware in these type of sentences when to use word "being" in Sr. No. 3, 4 and 5 and when to use a simple gerund (Verb+ing, sending) like in Sr. No. 6.
As per my understanding, in Sr. no. 3,4 and 5, "being" is used because there is an adjective (like able, quite and taken) and to show a kind of action also, we have used "being". In Sr. No. 6, a simple Action (sending) has been potrayed., therefore, no "being" has been used.
2) In Sr. No. 1 to 3 "am" represent, being. Can you please tell me when to use sentence like I am a vegetarian and when to use being a vegetarian ? Because both the sentences are identical to each other.
3) In Sr. No. 4 to 5, does being represent one of its form(am....) like in Sr. Nos. 1 to 3).
I think my question is clear.
Originally Posted by Casiopea
The underlined portions function as objects.
 Would you mind being quiet for a moment?
 I look forward to being interviewed on the current affairs programme.
 She was afraid of being accused of a crime which she did not commit.
The objects above are headed by the noun "being". It's a gerund (a verbal noun) and it's modified by the adjectives "quiet", "interviewed" and "accused".
Here's a gerund test you can try:
 Would you mind DOING THIS?
 I look forward to DOING THIS.
 She was afraid of DOING THIS.
 Swimming is fun.
TEST: DOING THIS is fune. ("Swimming" is as a gerund)
 Being here on time is important.
TEST: DOING THIS on time is important. ("Being here" is a gerund phrase)