The shmup, or the shoot'em up, is the oldest well-defined game genre. Dating all the way back to Space Invaders, Tempest or River Raid, the principle is simple: you're controlling a spaceship (or a plane, or a magical girl) and shooting at things that come your direction while dodging whatever they fire in your general direction. Do games get simpler than that?
I have a special place in my heart for shmups because of all the time I've spent with them as a kid, starting with the legendary Raptor. The shmup, together with the platformer, helped define my concept of entertainment. It also taught me the imporant truth that games can be hard... and that it's okay if they are. (Starting over is fun.)
Shmups are entertaining for very good reasons. First of all, they are immediately obvious. This is you! You can move in these directions. Shoot things, don't get hit. Secondly: Destroying numerous enemies is rewarding on its own. This is often underlined by polished animations of explosion. Add the fact that dodging enemy fire requires you to constantly use one of a human's most basic skills: predicting motion of multiple items at once. Sum it all up and you will get an efficient dopamine-inducing machine.
The shmup has been there long enough to develop subgenres:
Bullet hell (danmaku)
From DonPachi to Touhou, some shmups take the genre's selling points to the extreme. You like dodging? Here's a lot of things to dodge, enjoy! You like shooting? Here's a big gun that wastes everything in its path. You like bosses? Wonderful, you'll spend quite some time learning each one.
A danmaku shooter is all about dodging and predictions. Your enemies will shoot countless bullets, all aligned in elaborate, aesthetic shapes. You will learn these shapes and find the optimal path around them. No matter how big your player sprite is, you will only care about bullets hitting the very center of the sprite. This allows you to dodge precisely and squeeze in between bullets. Most shmups will give you a dedicated key for slow movement; some will encourage taking risks and give bonus points for grazing bullets.
Oh, and you die from a single hit. What would be the point of dodging otherwise?
The graphics come in various flavours, but expect a lot of symbolism - a danmaku shmup cannot be playable unless all incoming bullets are extremely visible. Everything else - such as enemy sprites and your own bullets - is secondary
Danmaku shooters can be perceived as not very accessible for new players, but titles like Jamestown go a great deal to make the genre enjoyable for everyone.
This funny term is sometimes used in a derogatory manner by danmaku fans. I don't think there's a strict definition of what makes a shmup "euro" (the etymology doesn't help). They way I understand it, the distinction is rooted in the game designer's fundamental vision of what should be fun about the game in the first place. If you ask a creative director behind an euro-shmup, he might brag about an immersive story, a variety of well-designed enemies, or perhaps about a complex system of upgrades. An author of a bullet hell would instead talk about bullet patterns and ingenious gameplay mechanics of dodging.
Euro-shmups tend to offer more casual gameplay. They are more forgiving - you can take a decent beating before having to restart. Most of the fun comes from actually playing the game and discovering the content it offers. There's difficulty, but there is less focus on requiring you to do superhuman feats of dexterity. However, this normally comes with a more boring difficulty curve and lower skill ceiling.
The role of pickups and power-ups is usually emphasised. While bullet hells usually take the minimal route (score-only pickups or basicmost weapon upgrades), euro-shmups like to add a lot of variety here. Pickups are how you would acquire better weapons, sub-weapons, rear weapons, weapon platforms, secret weapons, and superpowers. Or maybe the pickups are cash and there's a shop that you can browse between levels? Or many different shops...
This subgenre is more casual and less sporty than the previous one. but personally I don't have a single problem with it. The titles that introduced me to the whole shmup thing - Raptor and Tyrian - are also the most notorious euro-shmups.
The basic premise of "you shoot and dodge things" can be extended in a thousand directions, but there's a good variety of clear trends. With the advent of 6th-generation consoles, the shape of the standard controller has finally converged to a device that has the minimum of two (2) analog sticks. Let's do the simplest thing possible: the left analog is for dodging, the right is for aiming. Voila. You get a twin-stick shooter, such as Geometry Wars or Scoregasm.
Ditch the second analog, add a mouse - you'll get a common 2D shooter. Add character progression, get Crimsonland. Squeeze in a bunch of roguelite elements, get Enter the Gungeon. The variety never ends...
The shmup family is the oldest and noblest of all games. It has aged, but it's not going anywhere. Don't believe me? Go play some Genetos.