The Star Trek universe is largely populated with humanoid races - one head, two eyes, two arms, two legs – with relatively minor variations. One of the few non-humanoid, intelligent races we see on the show is the Sheliak in the episode "The Ensigns of Command" (TNG, season three). Their non-humanoid form creates an interesting backdrop for the even larger cultural and linguistic barriers that exist between the Sheliak and the humanoid races of the Federation. At the same time it poses some interesting questions about how they have advanced to become a powerful, space-faring civilization.
First it must be acknowledged that while the Sheliak are non-humanoid, they still have a roughly humanoid silhouette. What we see of the Sheliak is a human sized, upright being with a head-like protuberance at the top and undulating masses where a humanoid's arms would normally be. Still, they have no discernable face, eyes, nose nor any true appendages. Therein lies an interesting problem: How did the Sheliak become a technologically advanced society without any appendages?
In the Star Trek universe we have encountered intelligent non-humanoids without appendages in the past. The Horta is an obvious example. However, while the Horta is intelligent, it clearly is not capable of creating even simple tools let alone complex technology. We might also mention the jellyfish-like creature in the Voyager episode "Think Tank" who was incredibly intelligent, but only able to travel through space with the assistance of other species possessing both intelligence and the ability to create and employ technology.
Intelligence and technology clearly do not go hand in hand by necessity. No matter how knowledgeable a species might be, if they cannot manipulate their environment (if they don’t have opposable thumbs or some alien equivalent) they cannot create the technology to move beyond their own world.
So what of the Sheliak? In many respects they resemble upright cousins of the Horta, yet they have powerful technology at their disposal. How did they create it? How do they operate it? Here are two theories:
Perhaps they really do have appendages, we just never saw them. Throughout Picard's encounter with the Sheliak we never get a clear view of the base of the creature, where a humanoid’s feet would be. Nor do we see any outward sign of how the Sheliak director is able to interface with the systems on his ship, yet clearly he is able to do so. He can activate and deactivate communications, initiate a transporter beam (as when he "hangs up" on Troi and Picard), and when Picard quotes the treaty you can hear the Sheliak's computer accessing the relevant information for him.
All this being said we can speculate that at the creature’s base, obscured by the glowing rods that surround him, are appendages - perhaps arm-like or tentacle-like - that reach down into the floor of the ship, almost like roots of a tree. Perhaps these appendages also have sense organs on them, some sort of eyes, that allow the Sheliak Director to see what he is doing as he manipulates controls and consoles below him.
A second possibility is that the Sheliak have telekinetic abilities and therefore do not need physical appendages. They can simply interface with the ship functions and perform other tasks by thinking about it. Along similar lines they may have organs that allow them to manipulate electro-magnetic fields in precise ways, giving them control of metallic object in their environment. (Imagine a cross between a giant slug and Magneto from X-Men.) This might explain their apparent ability to interact directly with the electronics on their ship.
Whatever the method, the Sheliak must have some way to overcome their seeming lack of limbs, allowing them to become a technologically advanced and evidently an extremely powerful, if reclusive, force to be reckoned with.