A quick way to assess compatibility between two peoples charts are taking into consideration the four elements. The Earth signs are Taurus, Virgo, Capricorn. The Air signs are Gemini, Libra, Aquarius. The Fire signs are Aries, Leo, Sagittarius. And the Water signs are Cancer, Scorpio and Pisces.
The elements that suit each other best are Fire and Air or Earth and Water. Each element also has a great understanding with itself, such as a libra and a gemini, both air. Now when examining the chart you can compare the personal planets, Sun, Moon, Mercury, Venus, Mars to see how the two people will get along intimately on the day to day. If you want to see whether two people's future and life paths are aligned you can look at Uranus, Jupiter, Neptune, Pluto and the North Node.
For the quickest check I would compare moon signs since this is your truest inner selves. Look at Mercury signs to see if there is trouble communicating. And lastly I look at the north node to see if their destinies support each other.
Before you go checking all your friends and ex’s charts remember compatibility isn't everything. Checking compatibility is purely to see if two people will generally have a comfortable experience, but comfortability doesn’t birth growth between people, challenges do. Look out for my next article on what aspects in a chart are the best fertilizer for two people to blossom like a lotus in the mud.