Exercising the right to vote is a citizen's right because it is necessary that who and which government rules the country. As it ensures the development of country it is also very important for a citizen's development. Therefore all should vote for the right person as it is all about our future....
Right to Vote is a recognized right and not merely a duty under the Constitution of India, it is crucial to understand the difference between a right and a duty that we have as a citizen of India - or as a part of the democracy that is made by the people and for the people.
When you have/or are endowed with a right, you can enforce that right legally in the Court of law, if or when that right is violated or infringed upon. Certain rights are recognized by the Constitution of India (that cannot be legally taken away from you). For instance: Right to Equality (Article 14 in The Constitution Of India 1949), Right to Freedom (Article 19 in The Constitution Of India 1949), Right to Life and Liberty (Article 21 in The Constitution Of India 1949) among the other important ones. These rights to some extent, are also internationally recognizable by pertinent bodies like the United Nations.

Image Source: Career in Human Rights
When you have a duty, a sense of moral as well as societal obligation is attached to it. That being said, the Constitution of India lists down certain fundamental duties upon the citizens of India under Article 51A. These duties cover a range of areas like environment, education, as well as to promote peace, feeling of nationality, protection of national property etc.
So a right is obligatory as well as enforceable/recognizable by law, whereas a duty is not mandatory, but would be a positive action if one does follow them. Both rights and duties do interconnect at some point, which becomes relevant when you look at your conduct as a Citizen of the country. For instance: Duty to protect environment may relate to the Right to Life and Livelihood (which also includes living with dignity). Now that the concept of right and duty is little clear, understand that the ability to vote is neither a fundamental right nor a duty.