What is a crossword?

According to wikipedia, a crossword is a word puzzle that normally takes the form of a square or rectangular grid of white and shaded squares. The goal is to fill the white squares with letters, forming words or phrases, by solving clues which lead to the answers. In languages which are written left-to-right, the answer words and phrases are placed in the grid from left to right and from top to bottom. The shaded squares are used to separate the words or phrases.

Squares in which answers begin are usually numbered. The clues are then referred to by these numbers and a direction, for example, "4-Across" or "20-Down". At the end of the clue the total number of letters is sometimes given, depending on the style of puzzle and country of publication. Some crosswords will also indicate the number of words in a given answer.
Further and fuller details in this regard can be seen from the following link: http://en.wikipedia.org/wiki/Xword 

It’s a game, a game of fun & knowledge; crossword helps not only test but also to improve one’s vocabulary. This game stimulates, diverts, entertains and teaches millions across the globe. It helps pass (not kill) the time during travel, leisure, cozy companionship or in loneliness. There is no bar on the number, age, sex or qualifications of participants in any round of the game. "Word Square" the predecessor of present day crossword had existed for over two millennia, though in a much simpler form. In many parts of the world (rather developed world), this game has been the most popular indoor activity. 
Most newspapers of the advanced world publish these puzzles daily or weekly. 

The basic structure of a crossword puzzle, as most of you already know, is a square of any convenient size, divided into smaller blocks or squares. Some of these blocks are shaded while others are blank, for example: 

The clues to the above puzzle are given in the following fashion: 
1. Not in specified place physically or mentally. (6)
4. A beloved person. (4)
6. Main thing you are thinking about. (4)
7. Systematic plan of action. (6)

1. Nations joining together to fight a common enemy. (6)
2. A place within a region identified relative to a center or reference location. (4)
3. Aquatic and land reptile having a bony shell and flipper-like limbs for swimming. (6)
5. Condition of being comfortable or relieved. (4) 

The clues in the above fashion help us fill the blank blocks in the surrounding larger square. Some clues may be very simple giving a direct meaning of the word, whereas others might be cryptic, i.e. describing the word in an unusual way. At the end of each clue, a number in brackets refers to the size of the word, i.e. the number of letters forming the word. 
Based on the foregoing clues, the answers which come to the solvers mind are: 

Across: 1 Absent, 4 Dear, 6 Idea, 7 Scheme.
Down: 1 Allies, 2 Side, 3 Turtle, 5 Ease.
Based on these answers, the solved puzzle will look like this: