There is a number of ships, that use an ordinary cloaking device (in contrast to the Phase Cloak, which is a real specialty of the Shadow vessels). This special ability makes a ship or station invisible to ordinary sensors. Only ships or stations equipped with a Tachyon Detection Grid can uncover the presence of a cloaked ship. This can give a player a decisive advantage for surprise attacks and sneaking behind enemy lines.
That does not mean, that the ship is automatically impervious to attacks. A cloaked ship cannot fire its conventional weapons while cloaked. When firing at a target, the cloak is deactivated. Same goes for being fired upon while cloaked. And even most special weapons will automatically uncloak the ship when used. An uncovered ship under cloak is very vulnerable as its shields are down while cloaked. Activating the cloak takes 3 seconds to be fully effective. During that period of time a ship is also very vulnerable. Same goes for beaming in: Once the shields of a ship are down, any opponent can simply beam aboard without any obstacles. This may allow for capturing a ship, especially as there is no attack announcement, when being boarded. So such a beam-in attack may go totally unnoticed.
Although not draining special energy, once active, the activation of the cloak itself drains 100 special weapon energy. After that the special weapons energy stays constant. It is neither drained further, nor restored while being cloaked. But the cloak does indeed need special weapon energy, as once you run out of it, the ship will decloak momentarily. See als Romulan mission Call to Power. That initial drain of 100 special energy may already make the use of a special weapon impossible, if the remaining special weapons energy is not sufficient to use it. One example would be the Myotronic Inhibitor. It requires more special weapon energy than is left after even one activation of the cloak. An active cloak is depicted like this:
On the left hand side you see an uncloaked Vor'cha cruiser, while the right one is cloaked. You can clearly see it's slightly brighter colors and a slight transparency effect. Without a Tachyon Detection Grid the ship would simply be invisible.
Here is a list of ships, that do possess an ordinary cloaking device:
The Phoenix may not seem to be rightfully part of that list, but actually it is (see the description of the unit on the matter).
It may not seem obvious at first, but the cloak does not prevent an AI opponent from knowing where your units are, once they are cloaked. A cloaked unit will only be fired upon once it gets discovered or decloaks itself, but that does not keep the AI from following a unit around, that is supposed to be invisible for it. As a matter of fact the AI does not even need line of sight. The AI is aware of any unit placed anywhere on the map. It simply won't be able to attack a cloaked ship until it has a detector uncovering it. A similar effect happens for human players, when you turn of the fog of war in multiplayer or instant action games. Cloaked units will look like discovered cloaked units at any time, once they are not covered by shroud any more. Only the aspect whether you can target it depends on it's state of being discovered by a Tachyon Detection Grid.