You average SOLDIER members are mako-infused humans. You're enhanced, but still human. But then, what are those things? Their mako energy levels are exponentially higher than yours.
Makonoids first appear in Final Fantasy VII as mako mutants being held in containment units inside of the Nibelheim Reactor. Cloud Strife, Zack Fair and Sephiroth, in a flashback depicting events five years before the events of Final Fantasy VII, encounter the containment units while investigating the reactor, and discover Professor Hojo's experiments have transformed humans into Makonoids. The machines the Makonoids are inside of are actually designed to condense mako energy into Materia, but Hojo has modified them to instead expose a human to the condensed mako energy, mutating them.
Sephiroth begins to believe he was created by the same method as the Makonoids and this discovery is one of the elements that leaves him very unsettled for the rest of the mission in Nibelheim. Sephiroth then begins reading books found in the Shinra Mansion basement. After his research is finished, it becomes the final element that triggers his insanity. He then declares he is going to see "Mother", destroys Nibelheim, then heads for the Mt. Nibel Reactor where Jenova is being held.
No Makonoids are encountered at any other point in Final Fantasy VII, as an enemy or otherwise. They are, however, encountered in Crisis Core -Final Fantasy VII- as part of the Genesis army. The Crisis Core Complete Guide reveals they were created by the research of Genesis's group in the Banora Underground. Graffiti found in the holding cells tells the player who they were before they were transformed into Makonoids.
Makonoids appear in battle during the Hojo's Monsters mission. Another of Hojo's experiments, another Makonoid titled Experiment No. 121 has to be fought in the mission of the same name. In battle, Makonoids are strong, fast and unpredictable.
The most common Makonoid enemies fought in the game are the results of Project G. They are fought in the Banora Underground at Lake Oblivion and in various missions.