Render Line Trailing Player

This topic is so elusive to me that I'm not even sure how to title this question. Anyways, as with the game "Sunset OverDrive" where there's some line/texture following the player, or with the new game Jumpiter Jump on the iOS market, I often see things trailing the player. I'm pretty much just wondering how this is done? Documentation? Links to tutorials? Strange approaches? Black magic?

If anybody can shed any light on the topic, feel free to do so in this thread. Like I said, the concept of what it is is so elusive that I don't even know how to properly ask what it is I want to learn, much less where to go to learn it.

Here's an image to help you get the idea of what I'm looking to look into:

https://www.youtube.com/watch?v=kdUfidcECLs

That video has similar information, but I wouldn't want to instantiate and destroy particles over and over to achieve this effect, due to performance issues.