How much shade is the Sechach required to provide:
The Sechach must provide enough shade for there to be more shade than sunlight in the Sukkah. Thus even if the Sechach is very thin and thus allows sunlight to enter, so long as it provides majority shade to the Sukkah, and there is not one area of three Tefachim [24 cm.] without Sechach, then the entire Sukkah is valid. The definition of majority shade is that there is more area covered by Sechach than there is empty space.
Equal amount of empty space and Sechach: If however there is an equal amount of empty space and Sechach then the Sukkah is invalid. If however one sees an equal amount of sunlight and shade on the ground of the Sukkah then the Sukkah is valid.
The Chassidic meaning behind the requirement of majority shade:
The Sukkah represents the drawing of an Or Makif down below, to the Jew dwelling inside. Now an Or Makif is a very high level of G-dliness [level of Yashis Choshech Sisro] which can only be transmitted in a concealed fashion, hence the requirement for the Sukkah to retain majority shade.
A Sukkah that contains areas with more shade than sunlight and areas with more sunlight than shade:
If in total there is more shade than sunlight the entire Sukkah is Kosher, including the area that has minority shade, if the area with minority shade is not a size of 7 by 7 Tefachim [48×48 cm]. If the majority sunlight area is 7 by 7 Tefachim [48×48 cm] then although the majority shaded area is valid one may not sit under the area with majority sunlight.
Is there a maximum amount of Sechach that may be placed on the Sukkah?
Allowing the stars to be seen: Initially the Sechach must be thin and light enough for the large stars to be visible through the Sechach at night. Bedieved even if the Sechach is as thick as the roof of a house to the point that no rays of sun penetrate the Sechach, the Sukkah is valid so long as rain is able to penetrate through the Sechach as will be explained next. [The Chabad custom is to make a hole in the Sechach to allow one to see the stars.]
Allowing rain to penetrate: If the Sechach is thick to the point that even rain is unable to penetrate, it is invalid.
How much of the Sechach must be able to be penetrated by rain and allow star viewing?
Some Poskim rule that so long as the stars are visible from one area within the Sukkah it is initially valid. [Practically this is the Chabad custom.] However if rain cannot penetrate in an area of four Tefachim then that area is considered invalid Sechach for all purposes. Others however rule that so long as there is a 7×7 area that allows penetration of rain, the entire Sukkah is Kosher.
The Chassidic meaning behind allowing the stars to be seen:
The Sukkah represents the drawing of an Or Makif down below to the Jew dwelling inside. The purpose of this Or Makif is to be drawn internally into the Jew. This is represented by the stars which represent a glimmer of the Makif light.
 631/2; As the sunlight which penetrates the empty space in the roofing of the Sukkah spreads inside the Sukkah and hence on the floor of the Sukkah there is more sunlight than shade. [ibid] This indicates that the Sechach above is giving less shade than the air is giving sunlight and it is hence invalid. [Mendel Ben David]
 631/3; The reason why the Sukkah is valid when there is an equal amount of shade and sunlight below on the ground is because this indicates that the Sechach above is providing more shade than sunlight and this is considered complete Sechach. [ibid]
 As that area is nullified to the area that has majority shade. [ibid]
 The reason for this is because 7×7 is the size of a Sukkah and therefore it is considered a significant space which is not nullified or secondary to the part of the Sukkah that contains majority shade. [ibid]
 Hisvadyus 1990 1/164
 The reason: The reason for this is because if the Sechach prevents penetration of rain it is not considered a Sukkah, as a Sukkah is defined as a form of roofing that protects only from shade and not from other matters. However if it is also able to protect from rain then it is not considered a Sukkah but rather is similar to a house. [ibid]
 Peri Megadim
 Piskeiy Teshuvos 631/2