Like many editing tricks, you've probably seen J-cuts and L-cuts before without actually realizing that you were seeing a J-cut or an L-cut. Let's begin by explaining. In a J-cut, the sound of the next scene precedes the picture, and in an L-cut, the picture changes but the audio continues.
To understand how these cuts got their names, visualize the timeline in an editing program that is comprised of two consecutive clips. If your audio cuts from the first clip to the second clip, but the video from the first clip continues, it forms an J shape in the timeline. However, If the video from the second clip comes in before the audio for that clip does, this forms a L shape in the timeline. In ancient times, before "timelines" or digital editing even existed, these cuts were known as "video advance" and "audio advance." Here's a helpful graphic that illustrates these cuts:
Image courtesy of VideoMaker.com
Now let's discuss why you might want to use a J-cut or an L-cut in your project. Imagine you have a scene of dialogue between Person A and Person B where the clips cut back and forth every time each person speaks. In other words, you have a clip of Person A talking, then you cut to a clip of Person B talking, then you cut back to Person A when they begin talking again, and so on and so forth. This is referred to as a "ping-pong" editing, which can be disorienting and kind of stylistically boring. By using a J-cut, you could stay focused on Person A's face when Person B begins talking. Conversely, you might want to focus on Person B's face before Person A is finished speaking. In this case you would use an L-cut. In either scenario you allow yourself the opportunity to show the reaction of the person who is not speaking, which is a powerful tool provided by your actors that you should take advantage of.
Here are a couple examples of J-cuts and L-cuts:
Notice the J-cut at 0:22 when the audio from inside the capsule starts before you see the astronaut. This lets you know where the story is about to go and creates a smoother transition from one clip to the next.
Etsy's "Handmade Portraits: Liberty Vintage Motorcycles" starts with a short J-cut as a smoother transition and an introduction to the speaker, but uses an L-cut right after to show the shop as the audio continues.
*P***Pro tip:** In reality, while you're editing you probably won't be thinking "I should start a project that employs lots of J-cuts and L-cuts." As usual, the best editing is the editing that no one notices. The same applies to J-cuts and L-cuts. They should be used to subtly aid the flow of your videos and to keep your audience focused on the story you want to tell.