Been playing this recently.
This game is the 8th game in the series, and by
Team Shanghai Alice, a doujin circle that's actually just one person who calls himself ZUN.
This is basically one of those classic shooting games, now known as "shmup" in some circles. It's also called a danmaku ( 弾幕 ) type of shooter, or basically "curtain fire" as the game calls it. That's because there's often a *ton* of bullets on the screen so that the slightest mistake will kill you. And people revel in this sort of challenge. Just try it on the hardest difficulty, with no continues. I dare you.
Here's an example. Oh, and this was on the
easiest mode out of four difficulty levels.

This particular pattern is from one version of the final boss. You can't attack it at all, and instead, must survive 90 seconds of about 4 or so different patterns. BTW, I still haven't succeeded in perfecting it yet. My best of 27 tries was to reach 0 seconds on the timer, and then get hit by a bullet. -_-
What's also special is that these bosses have many many different attack patterns. Two per health bar. Specifically, their health bar is split into red and white. When it reaches the red, they use their spell card, which is basically their version of the player's bomb button. Not only that, to get the bonus, you must beat it the spell card perfectly. If you die or use a bomb yourself, you will fail the bonus. The extra stage is crazy, since the boss has 10 lives (so basically like 20 patterns to memorize). It's also not just a matter of memorizing their attack patterns too, since usually the enemies will target your current location on the screen.
It's also interesting that there's this new game called
Moedan, by the same people who created this book on learning English, to teach you how to play these games effectively. Or basically, danmaku shmup for beginners.
Here is a video by a guy who perfected the Last Words (probably the hardest boss patterns in the game). It's actually easier to do Last Words than to beat the game perfectly on Lunatic difficulty though, since you can retry immediately over and over until you do it right.