Very often we earn merit badges as a group. Some, like CIT Community and Sailing, are exponentially more fun in groups.
But say you are the only Scout who wants to earn it.
When deciding to pursue this, ask the Scoutmaster or any of the Assistants for a blue merit-badge card. (On a rare occasion, you may be declined; some merit badges have pre-requisites. A couple have physical requirements; for example, we will not put a 50-lb Scout on a 12-gauge shotgun.)
You will then receive a list of merit badge counselors to call. You call one, introduce yourself, and let him know you'd like to start said badge. Depending on the requirements, he may tell you to start and call him when it's done, or he may require multiple meetings. (For example, Pets requires a phone call from the Scout before starting the four-month term.)
Once you complete the badge (i.e., when the counselor signs you off, that you have completed the requirements), you give it to the Scoutmaster or to any of the Assistants. You will recieve your merit badge at an upcoming Troop meeting.