Have played Super Metroid? Or, say, Castlevania: Circle of the Moon? These are truly remarkable games! A Metroidvania is a game that resembles at least one of them. By inference, it must be great too! Are we done? I think we are! Thanks!
...are you still here? Okay, okay, I'll tell you some more.
You are a hero! What's your goal? Defeat someone evil? Find out what happened on an abandoned space station? Or maybe just find your way to safety? Maybe! Either way, there's a big area for you to explore. Be prepared: there will be a lot of combat along the way...
Your quest will change who you are: you will learn new abilities or discover items that grant you additional powers. Your combat skills are going to increase. Sooner or later you are sure to reach a dead end: you will have to navigate back to the areas you already know. There, you will discover that your new abilities allow you to take a new path that was not available to you previously. Likewise, the enemies that challenged you previously may now appear trivial to defeat - or maybe there was no way to defeat them previously, but now with your new abilities you finally stand a chance?
Your path through the game may appear linear at first, but the world is likely to open up more and more as you progress. In fact, exploring is encouraged from the beginning: there are hidden detours from from the main path. They are likely to contain additional challenges and rewards. If you want to arrive at the endgame prepared, you'd better explore as much as possible first! Your best shot is to remember as much of the scenery as possible: what appears unreachable now might actually be a new path available to you in the future as soon as you acquire new skills. By the time you reach the ending, you will be free to roam freely.
The Metroidvania can be seen as a specific flavour of the classic side-scrolling platformer, but the basic elements of gameplay that make it fun are more universal than that. The genre starts with Metroid, released for the NES in 1986. The side scroller genre has already been thriving around that time, but Metroid did a few new things: with little introduction, it placed you in the middle of an interesting world that you can progressively explore in order to discover more challenging areas and new abilities that help you deal with them. The anticipation of gaining a new power that would greatly increase your combat ability or unlock new areas for exploration proved to build a lot of immersion. Metroid topped it all up with fantastic art and varied combat using timed bombs and the iconic screw attack.
1986 was a fantastic year for NES games: the Legend of Zelda was released and became wildly successful. While it's not considered a Metroidvania, I'll risk a claim that it's a great game for the exact same reasons. Just see how much of the previous paragraphs applies!
The same year, the original Castlevania has been published. It started out as a linear platformer that involved little exploration and character growth, but excelled in enemy design and very tactical combat, amplified by the limited mobility of the protagonist (every jump needed to be calculated) and a range of weapons with limited number of uses. The series later attempted to explore the RPG territory (with somewhat amusing results) but only flourished as late as 1997, when Koji Igarashi married it with the best aspects of Metroid and added some RPG character development elements, giving form to Castlevania: Symphony of the Night.
Castlevania and Metroid are like two strands of the same line of DNA. They continue to manifest in various proportions in wonderful and well-received titles (most recently Axiom Verge. Let's also remember the moment when all the genre fans assembled on Kickstarter to help IGA himself fund his new project, Bloodstained: Ritual of the Night, and managed to raise $5.5M (that's 40% more than Pillars of Eternity that arguably had more marketing and also belongs to a more mainstream genre).
I like that metroidvanias come in great variety. Sometimes it's a full on Metroid tribute (hi Axiom), sometimes it's more towards old Castlevania (hello Odallus), and sometimes hey, someone just wants to prove that the genre can work in a minimalist setting too (Hero Core!).
Compare and contrast: Volgarr the Viking. This lovely game is NOT a metroidvania. Even though it appears to be greatly inspired by Castlevania combat, it drops the exploration in favor of a sequence of challenges that you're supposed to repeat and master.
Compare Dark Souls: if you ask me, that's pretty much the closest you can get to the genre in 3D. I'm not a huge fan myself, but I've seen the term "Soulsvania" here and there.