Familiars? Eh, well, as far as animals are concerned, such a bird of prey, it's the same as on Earth. If you can tame, train and/or befriend them, they're (more) likely to stick around respond to commands they have learned. There is no particular Dungeons & Dragons-esque magical bond between mortal species like that. (EDIT: Short of actual magical mind control, that is.)

The only bond similar to those would be between summoners and immortals they summon, of which there generally are three types of bond:
The weakest one is that of an immortal that has already been subdued to the summoner once, and which the summoner can be relatively certain it will be able to dominate again to force it into obedience. The only condition to the formation of that kind of bond is that the summoner needs to learn the immortal's name, so that they might summon that particular immortal again rather than just a similar one. (Because it is relevant in this context: demons are named in the Devil's Tongue, and angels are named in the arcane language.)
The second one is that of an immortal and a summoner who have made a deal, bargaining among themselves to establish an agreement of sorts, the conditions of which may vary wildly. The immortal will then be summoned by name, as normal, but is liable to aid the summoner without the need for this to suppress the immortal's will first. This kind of bond is only as strong as the participants are reliable, of course, since either might betray the deal and potentially break the bond.
The last one is the most cliché one, to be honest, but is inevitable nevertheless: the bond of friendship. If an immortal accepts a summoner, they may freely aid them whenever summoned. These kinds of deals are rare among angels, and almost nonexistent among demons, but are generally regarded as the strongest, most dangerous bond possible. Summoners with such a bond have immortal familiars that can be called forth without concern for betrayal or having to control them, making them particularly dangerous.