The Arabic parts (or lots as they are more accurately and anciently called) are based on the idea that while each planet in your natal chart has meaning on its own, the blending of three planets in your chart can bring a new meaning to the three working as a unit.
The Arabic parts are a series of calculations between any 3 planets in your natal chart. The entire natal chart is represented by a wheel, which is 360 degrees. Since there are 12 signs, each sign is 30 degrees. Any planet that is in your natal chart is also in a specific degree in your chart. For instance, you can say that Mars is in Aries or you can be more specific and say Mars is at 14 degrees of Aries (that means it’s in about the middle of the sign). It's often important to know what degree a planet is at, because if a planet is close to the end or beginning of a house or it's close to (conjunct) another planet, the degrees can tell you just how close.
The idea with Arabic lots is that the combined influence of any three planets can be calculated in a certain way, by either adding or subtracting each planet’s position. The total that you arrive at represents a degree in your chart that is sensitive. When it is triggered by planets moving over it or coming into contact with it, it may activate it’s principle in your life more recognizably.
Mercury is in retrograde motion as I write this, so I wouldn't be surprised if I've lost you with my muddled explanation! Let's have an example right off to clear it up. The most popular part or lot is the Lot of Fortune. The location of the Lot of Fortune in your chart can tell you where you are most likely to experience luck or happiness, some form of good fortune often on the material plane, or at least, where the potential exists. The Lot of Fortune is found by calculating the difference between the Sun, Moon, and Ascendant. Of course, they have to further complicate it by having a different formula for day births and night births:
Day births (AM): Ascendant + Moon - Sun
Night births (PM): Ascendant + Sun - Moon
So, to speculate here, someone with a Lot of Fortune in Aries in the 7th house might indicate someone who has potential for good fortune coming through relationships of all kinds, especially with people that are of a strong and courageous nature. There are all kinds of other lots. The Lot of Spirit (dealing with soul and spiritual matters), the Lot of Love, the Lot of Success, etc. This is really only an introduction; there are a lot of parts and I suspect understanding how to utilize them to the depth that they can be understood is sophisticated.
For further reading
Calculating the Lot of Fortune with Chris Brennan (video)
The Arabic Parts in Astrology by Robert Zoller