The Combine Advisors are a race in the Combine to which the posthumans are enthralled. They seem to direct operations on behalf of the Combine (if they are not in direct control of it themselves).
The combine Advisors follow a unique life cycle. They start out helpless, requiring the aid of an incubation pod to survive. Eventually, as they age, they gain enough strength to leave incubation. They soon realize the power they have when they search for food, being half-starved being in the incubation pods for quite some time. Searching for food, they generally discover their power while chasing their target. Once the target gets to a certain distance, the advisor, feeling hopeless, will discover that it can pull the target to itself. They also discover their mind probe ability, which is the concussion effect you see in Half-Life 2: Episode One. They discover they can not only use the mind probe to read minds, but also discover they can use it to stun their prey. Once the advisor reaches a height in strength, these abilities become much stronger. Eventually they arrive at a point which they are able to... persuade... anyone to listen to them.
The advisors in Half-Life 2: Episode Two
In Half-Life 2: Episode Two the advisors appear "in the flesh".
The Advisors have the ability to manipulate objects in telekinetic way. They have been observed to levitate (they have no obvious locomotive limbs), crush metal barrels and human corpses, lift and throw objects and even tear open buildings.
They also seem to have an, at least, basic grasp of telepathy and are able to induce mental pain in humans. The mental pain seems to be a side effect of the mind probe.
Combine Advisors are sheathed in a sort of jumpsuit. They have apparently become so dependent on their technology that they would be unable to survive without it. Mounted on the jumpsuits are a pair of arms to allow the Advisors to have some degree of fine motor control. This implies that their telekinetic abilities are limited in ways that necessitate direct manipulation. The machinery on the suits was once speculated to provide antigravity functionality. This may not be the case, considering the Advisor's telekinetic ability.
It is also possible that, like most other combine creatures, this "jump suit" is built into the being. Notice striders are naturally made of a sort of metal, as well as gunships and dropships. This would hint that most combine organic material is similar to metal. This would mean that the advisor is probably just another metal-like creature.
At least some Advisors have a prehensile proboscus. This has been seen to be used to penetrate human skulls with a stabbing motion. The method was deliberate (twice witnessed), and implies intent. It has been suggested that the most likely explanation is they use this to read memories from humans. However, since in at least one case they do this to someone who seems to have been dead for some time (the corpse of the rebel that Alyx finds draped over the Advisor's life support control panel in the barn), such an interpretation lacks credibility.
According to the commentary in episode two, this is actually how they eat.