Imagine if you had a whole bunch of Lego blocks and you wanted to stack them up as neatly as possible. You'd probably stack them in rows, with one row on top of the other. That's kind of what a lamellar phase is like. Materials like soap or lipids (which you can think of as the building blocks for cell membranes) can stack up in neat rows, with one layer on top of the other. This is kind of like how bricks are stacked up when building a house.
These layers are called lamellae (lam-uh-lee), and they can be found in lots of different biological materials, like the membranes surrounding your cells or the layer of mucus that lines your eyes. Scientists think that these neat stacks of layers help these materials to do their jobs better.
So, a lamellar phase is really just a fancy way of saying that certain materials can stack up in nice, neat layers. It's kind of like how you stack your toys up when you're done playing with them, but on a much, much smaller scale.