The concept of a design solution is 'The Big Idea'
A concept statement briefly explains the visual direction you propose for the solution. It provides the foundation and basic rationale for later choices you make with regard to color, images, layout, typography, etc. A concept is the essence of the innovative solution to a design or marketing problem - its how you would describe the solution to someone else. Its the 'big idea', the 'Eureka Ah Ha' moment that designers live for.
A design studio produced a piece to recruit students into a college theater program - the concept was driven by "Hey, I've got a great idea - How about a box of soap, since you'll get your hands dirty in the Theater program." That was the basic concept - it was refined and clarified and led to a successful solution to the design problem - Boxes of soap (with actual soap inside) were printed with the necessary program recruitment information. Or, for another piece, A bag of grass seeds with the tag line, 'Help us plant the seed'. The concept statement for the old UPS logo might be something like: A symmetrical mark integrating a shield beneath a present with a bow on it. For a restaurant, A standing whimsical cartoon pig that is fat and jolly, holding a knife and fork. Or A simple country kitchen cast iron with stove smoke that forms the words, Smoky Joe's Barbecue Kitchen.
Writing a concept statement
Describe, preferably in one sentence, the idea that serves as the solution to the problem design. It should be thorough but it does not need to be too detailed. The concept statement does not address objectives, specific type faces, exact rendering style (other than to say, abstracted or realistic or stylized, etc.) or too much minutiae. You're just talking to someone and describing the concept. Ideally, you should address the primary driving element first - A whirling stein of beer flying through space with the foam forming letters that spell 'Joe's Pub'.
When you write a concept statement, avoid putting it in first person. A design solution should not be about what you want, feel, or believe but what will work effectively for the audience (your wants are somewhat irrelevant to the client.)
Try not to get bogged down with specific detailed descriptors, objectives, criteria, project assignments, etc. The concept statement is just the big idea.
What concepts are not
• Color is not an idea. Type is not an idea. Illustration, photography, paper, film, computer, magazines, are not ideas.
• Target audiences are not concepts.
• Objectives are not concepts. A concept statement should not read, To [do something] . . .
• Concept statements should not include the word 'I' as in I think . . .
The purpose of developing a very specific target audience is that it will help in your later decision-making. Design is about communicating a clear message - it is important and vital to know exactly who you are talking to. Not everyone is interested in the design problem - nothing you say or do (short of offering free stuff, money, or fame) will reach those people and its a waste of time and client's money to try. Addressing people's interests makes the TM more precise and will help you tailor the message.