In the chart above, the rainbow coloured band containing the twelve zodiac signs is the ecliptic. This is the band in the starry sky through which the Sun appears to move during the course of the year, in its apparent annual journey through the twelve zodiac signs. It is also the band in the sky within which all the planets of the solar system will move as their travel through their orbits.
When you look at a chart, you see the ring of the zodiac (the ecliptic) around the outside, with the sun and the planets placed within the segments of the chart-circle in their zodiac position.
In the celestial sphere next to the chart, the ecliptic is the pink line around which the planets are clustered, and you can see the zodiac signs marked on this pink line.
The green line in the celestial sphere represents the horizon for the location of birth, London in this example. The green area is that which is below the horizon, while the black area is that which is above the horizon. The point where the ecliptic crosses the horizon in the east is known as the Ascendant.
In the above figure the Sun, on the ecliptic, is just about to rise above the horizon. You can see this in the chart as the Sun on the Ascendant, and you can see this in the celestial sphere as the Sun on the border between the green and black areas of the sphere.
In the chart you can also see that Mars is in the first house, and below the horizon; and if you look at the celestial sphere you will see that Mars, near the ecliptic, is below the horizon in the green area.
Indeed, you can think of the chart as like a flat plate. Turn it on its side, so you can just see its rim, and then imagine sliding it into the celestial sphere along the pink line of the ecliptic. Now you can see the planets in the celestial sphere, on or near the ecliptic, some above the horizon, some below. Then, looking at the chart, you can see the same planets, in the arc of the 1st to 10th houses.
It thus becomes clear that the chart is a two dimensional "slice" of the celestial sphere.