A dreamspike is a large, spike-like piece of silvery metal that is topped with a large metal head and set with golden inlay. 
The ter'angreal seems to make a type of barrier both in Tel'aran'rhiod and in reality, though the purple dome it creates can only be seen in the world of dreams. The dome has a radius of about four leagues and, when active, is always present with the Dreamspike at its center. Touching the barrier in Tel'aran'rhiod drains the energy from a person and can likely kill them.
The barrier prevents instant transportation in or out of it in both worlds. In Tel'aran'rhiod, willing oneself to be on the opposite side of the barrier simply fails to work. In the waking world, Traveling is useless except for those who are allowed by the user of the Dreamspike; gateways of all forms simply fail to form. This allows the Dreamspike to be used to protect oneself from hostile channelers, who will be unable to Travel to the user; it can also be used to trap a channeler in place by disabling the ability to Travel.
The dreamspike is also mentioned to have the ability to protects the owner's dreams, though how this functions is never shown. It may protect the dreams of the one controlling the dreamspike specifically, or it may simply block interaction with the dreams of those who sleep within the protected area. In either case, however, it does not block those within the barrier from entering or using the World of Dreams.
A dreamspike's other unique property is its interaction with Tel'aran'rhiod. Though it is a physical object and is inside the waking world when inactive, when triggered it disappears from the waking world and plants itself in Tel'aran'rhiod.
A dreamspike appears to have a "key" associated with it, presumably a mental command code that can be used to disable or counteract that specific dreamspike.
Occurrences at the Black Tower suggest that the other dreamspike is posted there as no one in the area is able to create gateways. It is likely that Moridin gave the ter'angreal to Mazrim Taim to use. When Perrin heas of trouble in the Black Tower concerning Travelling, he enters Tel'aran'rhiod and finds the ter'angreal. Cyndane shows him how to disable the Dreamspike in alternative to destroying it.