I assume they mean the Patrolling Horror district bosses that you need to kill for the Horror of Horrors achievement.
https://en.uesp.net/wiki/Online:Patrolling_Horrors

msetten wrote: »

What does “Slaying a patrolling monster in the Imperial City streets” actually mean. Just any monster that roams the streets, or specially that high level boss that patrols the streets?

It's a specific single boss in each aboveground district.
Amoncrul — Nobles District (big skeleton)
Baron Thirsk — Nobles District
Glorgoloch the Destroyer — Arena District
Immolator Charr — Temple District (big ogrim)
King Khrogo — Arena District (big daedroth)
Lady Malygda — Arboretum (big harvester if I recall correctly)
Mazaluhad — Temple District
Nunatak — Memorial District
The Screeching Matron — Elven Gardens District
Volghass — Memorial District
Ysenda Resplendent — Arboretum
Zoal the Ever-Wakeful — Elven Gardens District (big watcher)

Keep an eye on the map, note carefully the flag changes that signal roving bands of enemy players. A fast mount speed plus the passives that reduce the range of aggro is great for blazing safely past a bunch of random daedra when all you want is to determine if the boss respawned or if you are rushing across several districts to the next target.

Put at least one easy to use single target skill on your bar. If you need a boss for the achievement and it happens to be in the middle of fighting another alliance, carefully creep up and use that single target skill - you won't get noticed by the other alliance as quickly if you aren't giving them damage. Then if you get caught, don't respawn immediately. Wait for them to finish the boss so you get your credit for the kill.