A Search Position is a given coordinate in the world (X, Y, and Z). Most Mobs are able to have one, including Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, Gastgebers, and Calamities.
When having a Search Position, these Mobs will attempt to pathfind towards it every 2 seconds unless they currently have a target (entity they're attacking). Once getting within 9 blocks of their desired Search Position, it will be cleared and they will stop pathfinding. If a Mob is given a new Search Position when it already has one, it will be overridden.
Interestingly, the way Search Positions function for Calamities is slightly different than other Mobs, though it's nearly identical. The only difference is the distance from Search Positions that Calamities will stop pathfinding and clear their Search Position, which is instead 8.94 blocks, effectively the same.
Mobs can be given Search Positions in multiple ways:
Linked Mobs (with a Search Position) have a 10% chance every tick to set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers within either their follow range or a 32 block range (whichever is lesser) to the location of the original Linked Spore Mob's Search Position, assuming any of the Mobs found don't already have a Search Position.
Evolved Death Search Alerts have a 20% chance to be created and will set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers within a 60x60x60 block range (centered on the location of death) of the Search Alert to where the Evolved died.
Vanguards will set the Search Position of any Mobs they summon to the Search Position of the Vanguard that summoned them.
Broken Golems, every 10 seconds, will set the Search Position of any Infected, Evolved, Experiments, Bairns, Scampers, and Gastgebers within an 8 block range to the location of the Broken Golem.
Ringer Vigil Variants will set the Search Position of any Mobs they teleport to the location of the Ringer Vigil's current target.
Wombs have a 1% chance every tick to set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers within a 50 block range to where the Womb that set it is located.
Calamities (with a Search Position) have a 1% chance every tick to set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers within a 32 block range to the location of the Calamity's Search Position, assuming any of the Mobs found don't already have a Search Position.
Calamities, upon being given an Objective, will have their Search Position set to the location of the Objective.
Biomass Lumps, every 2 seconds, will set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers with at least 2 Kill Points within a 33x33x33 block area to the location of the Biomass Lump.
Reconstructed Minds, every 4 seconds, will set the Search Position of any Infected, Evolved, Hyper Evolved, Experiments, Bairns, Scampers, and Gastgebers with at least 2 Kill Points within a 50x50x50 block area to the location of the Reconstructed Mind.
The icon for Search Positions.
Trivia
Unlike other Systems, Search Positions aren't contained within one class in the code, but over multiple. This is also why Search Positions for Calamities function slightly differently.