Am I eligibility to apply for the Indian Army after 10th??? What the basic requirements to apply????????

No, you are not eligible to apply for the Indian Army after 10th, the basic requirements are 12th to apply and you need to appear in the National Defence Academy after 12th. This examination is conducted by the UPSC. Here are the details of this examination:

Nationality:
a citizen of India, or
a subject of Bhutan, or
a subject of Nepal, or
a Tibetan refugee who came over to India before the 1st January, 1962 with the intention of permanently settling in India, or
a person of Indian origin who has migrated from Pakistan, Burma, Sri Lanka and East African Countries of Kenya, Uganda, the United Republic of Tanzania, Zambia, Malawi, Zaire and Ethiopia and Vietnam with the intention of permanently settling in India.

Eligibility for army:
The applicant should be 12th Class pass of the 10+2 pattern of School Education or equivalent examination conducted by a State Education Board or a University.

Important dates:
Date of notification: 21 JUNE 2014
Last date of form submission: 21 JULY 2014
Date of NDA exam is 28th SEPTEMBER 2014

Physical requirements: