An Egglocke is a type of Challenge like Nuzclocke
It is mostly done by YouTubers
You may only catch the first Pokemon you encounter in each new area you explore. Fail to catch it and you get nothing for the area.
Every caught Pokemon needs to be swapped out for an egg
The eggs will be provided by other people like Subscribers and Friends
There is also an option for a Dupe Clause
A Dupe Clause is when a Pokemon is hatched the hatched Pokemon doesn't count if you already have that Pokemon and you can select another egg
Nickname every Pokemon you hatch
A Pokemon that faints must sent a Graveyard or RIP Box or Released.
A fainted Pokemon can never be used again
You must ditch your starter :(
A Few Examples of an Egglocke :
Here and This