Alien beings who infiltrated the highest levels of  Starfleet Command in 2364 on Stardate 41775.5.  Admiral Norah Satie, Admiral Gregory Quinn, and Captain Walker Keel of the starship Horatio helped uncover this conspiracy which nearly resulted in the takeover of the Federation.  These beings were responsible for the destruction of starship Horatio, and compromised Admiral Savar, Admiral Quinn, Lieutenant Commander Dexter Remmick, Captain Tryla Scott of starship Renegade, and Admiral Aaron. (“Conspiracy” TNG)  These beings were originally discovered by a Federation survey team on an uncharted planet in 2363.  These beings consider themselves a superior form of life humanoids.  These parasitic beings invade the human body, attaching themselves to the spinal column and base of the brain.  It takes complete control of all brain functions and breathes through a small gil protuding from the base of the victim’s neck.  By the placement of the tendrils to the victim, the parasite appears to stimulate the victims adrenal glands, generating great strength, and resistance to even heavy stun phaser fire.  These beings are extremely difficult to remove from a victim, and any attempt may likely result in death for the victim.  All phasers must be set to kill in order to neutralize a compromised individual.  During their invasion of the Federation in 2364, it was found that patience was one  of their virtues.  When Enterprise Captain Picard suspected their conspiracy, they did not go after him or his ship, rather they allowed him to come after them.  The one thing that both species, human and parasite have in common is a love for theatre.  They moved slowly and subtly to carry out their plans to conquer the Federation.  From their perspective, they mean humanity no harm, desiring a “peaceful coexistence” with humanoid hosts.  Apparently, these life forms cannot survive without their mother creature.  After their attempt to conquer the Federation in 2364, they sent a homing beacon from Earth to an uncharted sector of the galaxy.  The signal was aimed at an unknown region of the Menkare Expanse.   Several ships, including the Niobe and the Maximillian were sent to investigate.  However, after no contact other than a brief run in with an infested ship for over two years, the Federation determined that if there was a main staging area it was not the Menkare Expanse.

Further information is found at memory alpha:

• The neural parasites have thus far never reappeared on-screen, but have been shown in a comic book taking control of a spacefaring species called the "Onglaatu" in a DC TNG story entitled "The Broken Moon!".
• The parasites have also returned in Pocket Books' new DS9 series of novels. Several books in the "Mission: Gamma" series leading up to the novel Unity have revealed that the parasites are closely related to the Trill, and join with a host body in very much the same manner as a Trill symbiont does. They also reveal the Trill symbionts and the neural parasites have been fighting a long secret war, with several species worth of hosts and governments as their weapons against each other. The parasites' latest gambit has been the continued fervor for Bajor to join the Federation. For unknown reasons this would represent a great victory to their secret plan.
• In the novel Worlds of Star Trek: Deep Space Nine, Volume 2, Trill & Bajor by Andy Mangels and Michael A. Martin, the parasites are revealed to actually be genetically modified Trill symbionts, created on the remote Trill colony of Kurl. A deadly plague had been killing symbionts, and the experiments were intended to develop a cure. However, the experiment failed, and the symbionts so affected became violent and xenophobic - the parasites - and swore revenge on Trill society for this disaster.

What the  few parasites in the expanse and the original invasion were actually planning remains a mystery as it seems unconnected to their goals with the Trill.