Pursue is a command that causes your character to follow another without their consent. The other character may choose to lose you. This will match your adventuring.movement.following.pursuit against their adventuring.movement.following.evade. Flying away on a broomstick or using Divine Hand will also break pursuit, as can using skillchecked or restricted exits (for example, exits into worshipper-only areas of temples, or Order rooms).
It is also possible to suppress your ability to evade and lead on someone with the setting:
options combat suppress_evade=on
Those who are being led in this fashion get skill checks to notice this.
NPCs who use the pursuit command are fairly rare. Some--like cheetahs--only pursue if you attack them, while others--like Coffee Nostra hitmen, lynxes, yeti, and ice giants--pursue and then attack you. They may stop pursuing after some time has passed. Watchmen will also pursue, but even Captain Angua won't actually kill you.
Some npcs use another form of pursuit when in combat. This isn't the same as the command, and they don't follow you into the room immediately--instead, they enter the room they saw you go into after several seconds.
Many npcs in cities use this.