There are 3 main requirements to tame ANY familiar in Riders of Icarus:
1)Taming Points - each needs its own amount, for example, Elite Al-Shai requires 15 taming points, or Heroic Agnas The Red needs 40.
2)Taming Mark or unique item - some of the elite familiars and all heroic and legendary ones require some item to tame them like Dragon Emblem for taming Agnas The Red.
3)HP decrease - all heroic and legendary familiars have to be damaged enough for taming to be available(about 2-10% of hp has to be left)
Types of Familiars
There are 2 main types:
And there are 4 grades of those familiars:
In this short video you can look up how to tame all kinds of familiars and how its done.
To do this I reccomend you fly above groud types and jump on it without it noticing you.
For aggro familiars its much more trickier especially considering there are quite a few flying ones and those in dungeons, but you can get this bonus button, you will need to reset mount and while its running back to its spot go ahead and jump it while its running past you.