"Metroidvania" is a term used to describe a sort of exploratory 2D platforming game. It comes from the original games of the Metroid series and some of the middle-era of Castlevania games (most notably "Symphony of the Night", which is what primarily gave birth to the term).
The most defining feature is usually the world. Rather than have set stages or a world map, the entire game takes place inside one giant map, which you must traverse about. Not just any map, but typically a very complex map filled with all kinds of secrets. The map is typically divided into sectors, more in an aesthetic fashion to denote that the denizens and dangers will be different. Sometimes you have teleports to assist you in travel, but other times you just have to remember where to go in the map.
There's various elements at play in the exploration. There's typically tidbits hidden in spots, or puzzles that have to be returned to when you find appropriate equipment. The exploration is also non-linear - your limits of progression through the map are by various tools or powerups that you need to acquire, but the order in which you must acquire some of these tools is not set. And in many choices, there are many paths to get those tools, or even get past obstacles without those tools.
A very popular element of these games, especially in the Metroid franchise, is the concept of "sequence breaking". Using special tricks or otherwise mastering the utility of your basic abilities, you can sometimes access items and areas far earlier than the game expects you to.
As far as I know, Metroidvania games have always been 2D platforming games. Aside from the elements of exploration, the actual gameplay of Metroidvania games vary widely.
Aggro as a term originated in MMO's, but its usage has spread considerably.
Originally it was coined to describe any creature who would attack you on sight. So an "aggro" mob was one who would attack without being provoked, as opposed to one who wouldn't attack unless you attacked it first. The related usages here are "aggro range" or "aggro radius" which is the distance at which the mob will attack, and "aggro chain" which is whether or not the mob will bring his friends along, even if they are outside of normal aggro range.
As the games evolved "aggro" became the state of being attacked. If you were being attacked, you "had aggro", and if you did too much damage to something and it started attacking you instead of someone else you "stole aggro". If you did something stupid you could have "too much aggro" which meant too many things were drawn to attack you.
In your example, he's saying that the unit AI can be exploited because they don't chain aggro, so you can move into their aggro range, and individually "pull" them from their groups and eliminate them one at a time.
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Shopping is a colloquial term for a raid/attack in Clash of Clans. Generally if they say it's a "nice shopping trip", then they either got a lot of the type of loot they were looking for, or just a lot of loot in general. So in the replay where his defense won, the attacker probably got a decent amount of loot from the attack, and "going shopping" would be attacking some one else for loot.
Note: I have not heard this used by many people, and most of the people I have heard use it are from the UK, so it is possible that it is a regional term to use instead of raiding/looting/farming/attacking.