The Fused Shadows are the embodiment of the most powerful magic of Midna's tribe. Before they were banished to the Twilight Realm, the Twili sought to use their magic to take control of the Sacred Realm — this caused the goddesses to cast them into the shadow and forbid them to return to the world of light. The Fused Shadows appear to have been scattered across Hyrule, although Midna wears one as a crown.
When Zant uses Ganondorf's power to instate himself as false King of the Twili and casts the world of Hyrule into Twilight, Midna persuades Link to help her seek out the Fused Shadows, which are in a way her heritage, so that she may use them to defeat Zant. They are warned by the light spirit Lanayru that they are very dark and very powerful, and she and Link witness this in the way they corrupt those who come into contact with them.
But, when confronted with this power for the first time, Zant defeats them, curses Link "permanently" into wolf form and almost kills Midna by exposing her to the full brilliance of a light spirit. It is using the power of the Fused Shadow in anger — anger seems to be key — that Midna finally defeats him, and she also uses it to break down the barrier around Hyrule Castle protecting Ganondorf.
The Mirror of Twilight is the only link left between the Twilight Realm and Hyrule by the goddesses when they banished the Twili there. It is nigh-indestructable and can only be shattered completely by the true leader of the Twili — this is demonstrated when the false King Zant tried to break it in order to prevent Link and Midna from following him into the Twilight Realm and only managed to break and scatter it.
As Link and Midna see, the Mirror of Twilight is also very powerful. Yeta, a kind of yeti creature living in the snowy North of Hyrule, becomes so obsessed with her mirror fragment, although it's making her sick, that when Link and Midna try to take it she attacks and attempts to kill them. When they finally take it from her, Yeto, her husband, tells her that her reflection in his eyes is all she needs and she is contented.
Because of the demonstrations of this power, Midna muses on the nature of the Mirror of Twilight, even saying that, though they need it to reach the Twilight Realm, "[they] could be assembling something truly terrible here [...] something that [they]'ll ultimately have to destroy". I think it is this belief which led her to shatter the Mirror of Twilight as she returned to the Twilight Realm, even knowing that she could never see Link or Zelda again, with a bittersweet, "[A]s long as that mirror's around, we could meet again".