alright here's a quick run down on the most basic way to read a chart.
so the easiest way to get an idea of what your birth chart says about you is to look at what planet is in what sign, I'll post an example.
I'm using George Orwell's chart in this example because it's remarkably similar to my own and it's one I'm pretty familiar with...
We are all familiar with sun signs from the horoscopes in the daily news and the ever-present old pickup line "hey whats your sign?"
As you can see in the chart old george's sun is in the sign cancer, which means mr. orwell would be reading the cancer horoscope in our newspaper. The sun represents your most inner self, who your really are deep inside. So on his most deep inner lever he is a very intuitive, kind, caring, but also moody and somewhat stubborn person looking at the meaning for the sun and the meaning for the sign cancer.
Next we look at the ascendent (also known as the rising sign) this is the sign that was coming over the horizon at the time of birth and why having an exact birth time is so important. In this chart you'll notice the little -ac at the very left of the circle is in the sign virgo. The rising sign indicates ones personality and how they present themselves to the world. So looking at the characteristics for virgo its very likely george's friends would describe him as a person who is very analytical and observant but also tends to be inflexible and skeptical a lot of the time.
The third most important is the moon sign. As ragabr said the moon represents our unconscious desires and drives, it's our emotional self. Old george's moon is also in cancer (this means he's a double cancer) and thus emotionally he is very loyal and caring and can be quite moody, clingy, and oversensitive.
Given those three signs you can get a pretty good overview of a person's personality and how they relate to the world. now we'll do a couple planets and give ya an idea of how those show you different aspects of your life path.
The first planet we'll look at is mercury, ragabr described it very well as representing the symbolic ordering aspect of ourselves, mercury is the planet of our mind and mental sphere. George's mercury is in gemini (this is especially powerful as mercury is the ruler of gemini thus this is an exhaulted position for mercury). a quick look at the characteristics of gemini show us that mentally he is very clever imaginiative and quick-witted, but on the other hand can be somewhat superficial, impulsive, and restless gemini is the sign of the twins and is oft described as being two faced.
The last planet we'll look at is venus before we send you out on your own for the rest of em, this planet represents love and romance in our lives. George's venus is in leo, thus in the realm of romance george can be confident, ambitious, and loyal but on the reverse is stubborn, vain, and kind of pretentious at times.
and on from there mars is physical energy, jupiter is related to expansion and higher learning, saturn has to do with control issues etc...
after determining the planet and sign positions we look into the houses. The houses start just below the ascendent line and run counter clockwise around the circle.
The first house is the house of the self and usually relates to childhood and how you raised, we see in george's chart this house is ruled by the sign libra. This means goerge was raised up to be a very diplomatic, graceful and peaceful person but also learned to be very indecisive and superficial and unreliable.
the second house represents the inner and outer resources we have available to us, classically speaking this is the house of desires and obligations. In george's case this house is ruled by scorpio. Thus george has a lot of passion, is resourceful, and is dynamic or takes well to change, on the flip side he also has the ability to be very manipulative and can be very suspicious and jealous.
and this goes on around the circle, third house is about community, fourth is our emotional foundation and unconscious self image, fifth house has to do with love and emotional release, etc....
Wow that's a lot of stuff to take in is it not? Once you've gone over all that you have the basic foundation of what a person is about, but we've only barely scratched the surface. From here you get into the real astrology, you look at the mathematical angle's the planets make and how they relate to one another in the chart, then you can get into comparison charts or synastries between two different people and how they relate on the basic level of planets houses and signs and the more advanced levels of mathematical relationships, you can also make predictions for the future by doing synastry charts by overlaying the chart for a specific time period on a persons birth chart and noting the various relationships of the planets,houses, and signs.
ha and you thought astrology was just some vague forecasts in the the entertainment section of the newspaper.
There is tons of info online if your interested in learning more, I highly suggest you at least look up the meanings for all the planets houses and signs to get a good idea of what your chart has to say about you.
Hope this helped ya understand how to read your chart a little better.
-namaste