Before one can apply for a J-1 visa, he/she must be accepted for one of the Exchange Visitor Program categories and must have a sponsoring organization. The applicant also has to show a consular officer that he/she will remain in the U.S, only for a limited, specific period of time, has enough funds to cover all expenses in the U.S., and that he/she has binding ties with his/her country of residence, insuring they will return to it.

In order to obtain a J-1 Visa, the applicant needs to apply for the visa at an Embassy or Consulate.  Part of the application process includes an interview at the embassy consular section for applicants between the ages of 14 and 79.  An interview appointment and visa wait time varies for each country, but can be found on the State Department website.  

The procedure for obtaining a J-1 Exchange Visitor Visa is as follows:

1. Submit Form DS-2019 Certificate of Eligibility for Exchange Visitor Status. This form will be a Student and Exchange Visitor Information System (SEVIS) generated Form DS-2019 that your sponsor should have provided you with. 
2. Submit Form  DS-156 Nonimmigrant Visa Application
3. Submit Form DS-157 Supplemental Nonimmigrant Visa Application, which provides information regarding your travel plans. Note: All male applicants between 16-45, as well as applicants from state sponsors of terrorism (North Korea, Cuba, Syria, Sudan, Iran, and Libya) over the age of 16 must complete this form.
4. Submit Form DS-158 Contact Information and Work History
5. A passport valid for travel to the US, which will be valid six months beyond the end date of the J-1 visa.
6. One 2x2 photograph
7. Non-refundable application processing fee of 100 US dollars.

Once the J-1 Visa is approved, there may be an additional fee charged, the visa issuance reciprocity fee.  This fee will only be charged if the country you are residing (i.e. the country you are living in when you apply for your J-1 visa) in charges U.S. citizens for a similar type of visa.

For example the United Arab Emirates (UAE) charges $52 for a U.S. citizen to get a J-1 Visa to the United Arab Emirates.  Therefore if a citizen of the UAE receives a J-1 visa to the U.S., he/she will have to a visa issuance reciprocity fee of $52.

For more information about J-1 visa, please refer to the following links:
Description
Work for Non-Sponsor Employer
Duration
Extension
Benefits
Limitations
Procedure to Get a J-1 Visa
J-1 Waiver
No Objection Waiver
Independent Government Agency Waiver
Independent Government Agency Waiver for Physicians
IGA Listing
Exceptional Hardship Waiver
Fear of Persecution Waiver
State Health Agency Waiver (For Physicians only)
State Health Agency Listing
Green Card
Services that We Provide
Frequently Asked Questions about J-1/J-2 Visa
Frequently Asked Questions about J-1 two-year rule
Frequently Asked Questions about J-1 Waiver
Attorney's Fees
Articles & News on J-1