e.g. The bar graph below shows the TV viewing habits of Koreans.

In this sentence, what is the part of speech of 'below'?

Can we write like this?: The bar graph which is below shows the TV viewing habits of Koreans.

In the first sentence, "below" is being used as an adjective, telling us about the bar graph.http://www.uottawa.ca/academic/arts/writcent/hypergrammar/adjectve.html

You can write it as you have in the second version, but that sentence is considered "wordy." (Too many words to say something simpler.) The first sentence is better.
http://grammar.ccc.commnet.edu/grammar/concise.htm

=)

In the sentence "The bar graph below shows the TV viewing habits of Koreans," the word "below" is functioning as an adverb modifying the verb "shows." It provides information about the location of the bar graph.

Regarding your second question, yes, it is grammatically correct to write "The bar graph which is below shows the TV viewing habits of Koreans." In this sentence, "below" is used as an adjective modifying the noun "graph," and "which is below" is a relative clause providing additional information about the specific bar graph being referred to.