Layered animation generally refers to the idea of blocking in one part or section of the body at a time. One example would be animating the up/down of the hips in a walk first, and nailing down that timing since it's going to affect every other aspect of the walk. Once you have that, you could then do another "layer" of animation by animating the torso of the character. Maybe then you'd do the feet. Then the arms and wrists. Then the head.
All of those layers will combine to form one walk.
The trick here is to do it in such a way that the overall finished walk feels like ONE character. You want all these different "layers" of animation to come together to form one cohesive action, with the different parts of the body being driven, pushed, pulled, and rotated by the other parts of the body.
Our bodies are incredibly inter-related and connected throughout. You cannot do a big and fast arm motion without moving your shoulder, chest, head, and probably your hips and other arm.
There is nothing wrong with the layered approach, but it requires a very deep understanding of the way the body works together (something I often refer to here as "body mechanics") in order to have a finished piece of animation that feels like one cohesive body. It requires careful planning, reference, observation, and a fair amount of adjusting the different layers to work properly each other.
Personally, I use a layered approach when diving into a scene with a lot of action. If a character needs to walk a long way, climb something, and then jump down - I'll probably use the Glenn McIntosh "hide the legs" approach, which helps me not get too distracted when I'm working in a layered way. (If you haven't read my article about hiding the legs, you can read it here and then come back!)
Basically, I'll create a new layer in Maya, add the legs to it, and hide that layer. Now I've got a floating legless character. I'll grab that character by the root or whatever it is that moves the character but not the legs, and animate just that thing through the whole scene. I'll work on just that root node until the general timing is worked out as far as where that character will be, when the general up/downs will be from the footsteps, etc.
Then I'll leave the legs hidden, and pose the upper body (torso, arms, head, etc), knowing in my mind or from my thumbnails how I'll eventually want the legs to work in that pose. Remember - this is all largely planned out ahead of time, and I'll know 100%, at least in my head, what that pose is exactly going to look like when it's finished. I'm not "exploring" at this phase, I've already made most of my animation decisions and am simply inputting them into the computer.
Last, I'll turn those legs on, and the leg animation is much easier than normal now, since their movement and timing is completely dictated by the body animation whose timing I already am happy with. If the leg is about to hyperextend, then that simply means it's time to lift it up off the ground and do a step. There will obviously be a bit of back-and-forth here, touching up the timing and placement of the hips to make sure everything is just right, but overall this method has served me well for action shots.
The whole point of the method above is to block in the part of the body that will define the movements of the rest of the body (the hips and torso), as well as whatever part of the body audience will see most - you do that FIRST, in the layered approach.
For any scene where the character stays generally in one area of the frame - smaller actions, acting, etc - in those scenes I'll do a pose-to-pose approach where I'm posing out the whole body and saving a key on every possible controller on those key poses and breakdown poses.
In my opinion, the layered approach is far more difficult for newer animators, and my advice to anyone who hasn't been professionally animating for a few years already would be to stick with the pose-to-pose approach, using heavy reference, research, thumbnails, etc.
However, I will say that for more advanced students or professional animators, the layered approach is worth trying to see if it's something you connect with. Be extra vigilant, however, that the end result doesn't feel like a bunch of disconnected body parts that have been animated individually! You want one cohesive character as your end result, and don't give up until you have it!