A "blue card," or officially "Application for Merit Badge" is a document that indicates a Scout's interest in a merit badge, and intent to earn a merit badge.
The Troop website has a great write-up on earning merit badges, how to find counselors, and the order work must be done in. Below is an example of how to fill-in a blue card. Please note, a correctly filled out blue card is the prerequisite to starting a badge!
In the images below, the sections are color-coded to indicate who is responsible for filling out which portion.
RED text is information the Scout is responsible for filling in. Fill in the information as appropriate to you.
BLUE text is the information your Scoutmaster is responsible for filling in. The Scout should not write anything in these fields.
GREEN text is the information your Merit Badge Counselor is responsible for filling in. The Scout should not write anything in these fields.
In order to start a badge, you must speak with your Scoutmaster and get a Blue Card, fill it out and get your Scoutmaster's signature. Your blue card should look similar to the images below when you give the blue card to the Merit Badge Counselor. Remember to fill in the information as appropriate to you, and meet with your counselor before starting work on the badge to talk about the requirements and the work you need to do!
When all the work for the badge is complete, your counselor will give you back a signed blue card. this is your record that the badge is complete! Bring this record to your Scoutmaster so he may record it in your advancement record. Do not lose this blue card, as it's the record of completing the badge.
When you finish your badge, your counselor will give back your blue card filled, it should like something like the images below! The counselor will keep the "Counselor's Record" portion of the card, and give you back the "Information for Applicant" and "Applicant's Record" portions of the blue card. Give these to your Scoutmaster for record-keeping!