The Tyranid Hivemind is the collective consciousness of the Tyranid race. It knows only one thing: a complete and insatiable hunger.
Official Canon Edit
Because every Tyranid is an appendage of the Hive Mind, its power is very vast. Its mere presence can blot out all warp communication, suggesting that it may even be more powerful than the Chaos Gods, though it is entirely single-minded in its purpose of consuming all organic life. Regardless, because of the absolutely simplicity of its desires its likely immune to Chaos corruption.
The Hive Mind constantly learns from its many battles, meaning that the same trick won't work twice against it. It will send out leader organisms called Synapse creatures to control the lesser Tyranids; without them, the Hivemind's presence weakens and they become disorganized.
TTS Canon Edit
The Emperor claims to have made psychic contact with the Hivemind, stating that "it is like talking to a herd of hungry sheep; It is unfulfilling, and makes you look like an idiot." Likewise, the Hivemind connected to the Chaos Gods' warpchat using its synaptic mind to tell them that they were going to eat the galaxy (before the connection dropped). The Hivemind also made contact with latent psyker Bruce Norring, in an image that absolutely horrified him to his core.