Passive Perception: the DM does not want the players to roll
Active Perception is not when a character 'actively looks' for hidden monsters, it is when the player actively makes a Perception roll (because the DM told them to).
Passive Perception is not an intuitive sense the characters have even when they don't search, it is when a character actively looks but the DM determines the result by using their passive Perception score without the player making a roll.
Any ability check, active or passive, is a result of the character's activity and conscious intent; as described in the PHB's section on Ability Checks (emphases mine):
An ability check tests a character's or monster's innate talent and training in an effort to overcome a challenge. The DM calls for an ability check when a character or monster attempts an action (other than an attack) that has a chance of failure.
Passive ability checks are included as a subsection of the Ability Check rules and must follow the general rules for ability checks; they happen when the character is making an effort for them to happen.
Further, while it can be assumed that most of the time your character is deliberately on the alert for danger, if they focus their attention on another activity, they lose the ability to make passive Perception checks, as described in the PHB section on Noticing Threats:
Use the passive Wisdom (Perception) scores of the characters to determine whether anyone in the group notices a hidden threat...Characters who turn their attention to other tasks as the group travels are not focused on watching for danger. These characters don’t contribute their passive Wisdom (Perception) scores to the group’s chance of noticing hidden threats. However, a character not watching for danger can do one of the following activities instead...
'Active' and 'passive' thus refer to what the player and DM are doing in the meta-game (rolling or not), but not what the characters are doing within the narrative. You should not be determining whether the players 'get' an active roll based on what they say their characters are doing, but rather on how you as a DM want the meta-game to proceed. What the players say the characters are doing is the basis for permitting a check in the first place, regardless of whether you decide it will be active or passive.
Unfortunately, the currently-accepted and most upvoted answer here is simply incorrect about this.
The purpose of passive checks is explained in the PHB:
A passive check is a special kind of ability check that doesn't involve any die rolls. Such a check...can be used when the DM wants to secretly determine whether the characters succeed at something without rolling dice, such as noticing a hidden monster.
The answers of mattdm here and enkryptor here are correct in that passive checks resolve the characters' active attempts to perceive their surroundings. However, their answers were written before Jeremy Crawford's tweets and interviews were made unofficial and are thus burdened by trying to take his advice into account. Crawford's tweets and interviews are no longer necessary to explain RAW and in this case contradict what is in the PHB and DMG.
The DM in the video cited by OP was faced with a number of characters who could have perceived the Hidden goblins. Three of these characters explicitly said that they were looking for threats - one was on lookout duty, and two were moving through the brush, searching. The DM could then have used their passive Perceptions against the goblins' Stealth, or could have called for them to roll. Simply put, the DM called for them to roll because in a situation in which either a passive or active check was possible, he chose to have them roll actively. Whenever a DM decides that a Perception check is called for, they also determine whether the check will be rolled (active) or not (passive).
As far as why he chose to run the encounter that way, we can only speculate. The PHB explains that passive checks are explicitly for when the DM does not want the players to know the results of their rolls - in the case of looking for Hidden monsters, the DM does not want a player to know whether their character failed to perceive something because there was nothing there, or because there was something there but they rolled low. In this particular case, there was a fourth character making for the dead horses who was about to spring the goblin's ambush. Since the goblins were about to reveal themselves regardless of whether or not they had been perceived, it seems plausible that the DM decided that there was no reason to keep the results of the players' checks hidden, and so had them make active rolls.