In the most simple terms, the natal astrological chart is simply a diagram of the Sun, Moon, and the planets at the exact time your were born as seen from the place of your birth.

That's it, just a diagram on a piece of paper (or a computer screen).

Different schools of astrology draw up very different diagrams, and indeed different astrologers from the same school of astrology often draw up different appearing diagrams, but the fact remains, it's just a diagram of the Sun, Moon and planets.

The reason all these diagrams look so different is that by the placement of the planets in the chart, different sorts of information can be emphasized or de-emphasized, which can be important depending on the branch of astrology. The diagram also includes the results of very sophisticated calculations showing mathematical relationships between the planets.

The complexity of the math often scares away potential students of astrology, which is too bad, because with the advent of computer programs (and some are definitely better than others) point and click is the only computational ability required today.

The interpretation of the chart is the part which requires long study on the part of the astrologer these days. Again different branches of astrology will place different nuances and perspectives on a particular part of a chart, and the training of the astrologer will reflect that.

And here's the answer to a trivia question: even identical twins have different natal charts. Not different by very much in some cases, but different enough that the contrasts in their personalities can be seen in the two charts.

A natal chart is unique. No two natal charts are identical, unless you get the very rare situation of two different mothers giving birth to two different babies in exactly the same location at the same time with the babies taking their first breathes at the same time. Twins don't do this for one is born a few minutes after the other.