The Kekkai (結界) are both man-made barriers which form a sort of spiritual network throughout the world as well as the Spirit Shields which the Seven Seals create in order to spare the material plane from damage when they do battle. As man-made structures, the barriers' spiritual network prevents most major natural catastrophes from destroying human civilization. Because of this, humanity is able to live (and continue to pollute and otherwise defile the Earth) in relative peace and prosperity. Tokyo's barriers act as the keystones to this network; it is the proverbial "tie that binds". Therefore, if the barriers in Tokyo were to be destroyed, eventually the entire worldwide spiritual network of barriers would come undone, and with it the protection they provide. Earthquakes would then wrack the planet, destroying humanity and allowing the Earth to recover from the damage which humans have done to it (this is the ultimate goal of the Dragons of Earth).
As the barrier fields created by the Dragons of Heaven, the Spirit Shields are fields of varying geometrical shape (depending on which of the Dragons of Heaven has created the Spirit Shield) which only those involved with the end of the world can exist within. The barrier field effectively creates an alternate plane or dimension resembling the area of the material plane which the Spirit Shield was created to protect. This ensures that (so long as the one who created the Spirit Shield remains alive, conscious, in relatively decent health, and maintains an undisclosed amount of concentration required to create the Spirit Shield in the first place) the area in which the Spirit Shield was created and all the ordinary people within it cannot be harmed. Also, as a general rule, it is impossible for even the Dragons of Heaven and Dragons of Earth themselves to leave the space of a Spirit Shield until the creator of the Spirit Shield releases it. Trying to leave a Spirit Shield is said to be extremely painful and dangerous, requiring a great deal of power to even attempt. The rules regarding entering a Spirit Shield, however, seem to be more obscure.
Another way to easily put it is that a Spirit Shield is like a carbon copy of the battlefield combatants can do battle without fear of hurting innocent people or any interference from the outside. If any damage at all occurs in the city inside the Spirit Shield and the one who puts it up lives or kills the other, then no damage will be done to the real location in the primary dimension—as if the battle had never happened. However, if the one who set up the Spirit Shield in the first place gets injured to the point that he or she cannot maintain the requisite concentration to support the Spirit Shield, falls unconscious, or killed altogether, then the Spirit Shield is destroyed, and the damage done to the location within the Spirit Shield during combat is done to the primary physical dimension, killing many innocent bystanders in the vicinity and causing massive destruction.
It should be noted that Spirit Shields can be erected only because the person who set them up have someone they want to protect, and if the person they wish to protect dies—then the Dragon of Heaven's ability to erect the Spirit Shield dies too.