This, in fact, is a very complex question. In India, there are many different types of visa, for example for tourists, businessmen, journalists, and researchers. International interns come to India on several different types of visa: Tourist Visa, Entry Visa, Employment Visa, Business Visa or Student Visa, none of them being applicable to all internship scenarios.
The Entry Visa (X Visa) is generally for people whose purpose of visit to India does not fit into any of the standard visa categories, such as people coming for short term courses, volunteer work, or unpaid internships to India. You should apply for an Entry Visa along with a copy of the offer letter stating the terms of your internship position. If you are a student and the internship is part of your studies you should also obtain a letter from your university explaining why you are going to India.
Internships with a “stipend”, allowance or salary are generally considered to be paid internships and these technically demand an Employment Visa (which requires additional documentation such as the employment contract and a proof from the employer that no suitable Indian national could be found to fill the position). However, many organisations might directly ask you to come on a Tourist Visa because it is easier for them and for the applicant – although it is technically not legal.
Before applying for any kind of visa make sure that your passport is valid for at least 6-18 months, depending on your specific visa. Applicants who are not citizens in their country but residents must send a copy of their residence permit with the application. It is always advisable to consult the respective embassy in your country, for example the Indian embassy in Spain, Germany, Italy etc. There you will find most of the information considering your visa.