Kundali is the Hindi for what we call 'Horoscope' in English. It is a pretty widely used thing in astrology. It is based on the fact that heavenly bodies and their positions at the time of an event effects how the event goes on.
So in the case of the birth of a child, we record when he/she is born, at what place and other details. Astrologers, then on the basis of position of prominent stars, 7 planets, Sun and the Moon about that time create a chart, similar to the pic you posted (number of charts is not limited to 1).
The numbers in the various sections indicate the position of different heavenly bodies. By position, I don't mean position in space, but position corresponding to a specific house in the Kundali (which ultimately depends on some calculations based on the time and spatial position)...
Not only on the birth of someone, but astrologers also use the Horoscope calculation in numerous other events. When sacred rituals take place, we try to do them in specific time spans called Shubh Mahurats which are considered good. These time spans are predicted the same way Horoscopes are calculated, but in some different context.
Shubh Mahuratas are calculated/predicted for most things like marriages, ring-ceremony, baby-showers etc.
Nowadays, you can encounter that people first concern Astrologers for these time spans and request doctors to have their babies delivered near that Shubh Mahurat so that life of the baby be good. (I know this is a bit weird, but it does happen, somewhat rarely though!)
There are total 12 houses1 in all. Each house can accommodate a heavenly body.
Each house also corresponds to specific areas of life. The first house, for example signifies life, longevity, self, health, nature and appearance of native; Complexion, vitality, sorrows, gains and profits to younger brothers and his friends. It governs head and face.
The heavenly body present in the house signifies how the person's life will go on in that area. These heavenly bodies keep changing their positions with time; thus, your daily horoscope changes. The change depends on which graha comes in.
Further detailed horoscope information for a particular person can be calculated for specific transits by astrologers, when given birth details.
This is done by seeing the most initial graha which was present in the house. Then, we calculate which graha will be more strong. If a strong (and good) graha comes in, life in that area will be good and vice versa.
1. Reference - Astrology Basics See this link for more details on houses.