Usage - Press the "C" key; in sector mode, cleaves sectors; in surface mode, cleaves surfaces; in edge mode, cleaves edges.
Cleaving is the basic tool in JED. The cleave tool is used to shape sectors into what you want them to be (i.e. - a square, triangle, octagon, etc.). Cleaving is an alternative to moving the sector or parts of the sector (i.e. - moving surfaces, edges, or vertices) which can cause errors especially when adjoining or merging.
Cleaving is also used for "cutting" holes in surfaces to join two sectors (rooms) by creating a hallway between them. Cleaving is also helpful for aligning or adding different textures to your level(s). You can also use the cleave tool to split an edge into two edges (though not often used).
Now, let's get started. Switch to sector mode and select a sector. Press "Shift + 1" to switch to top view and align the grid. Press "C" and you should see a "lightsaber" : ). Right-click and drag and you should see a red line. Drag the line across the sector and release the button. Try cleaving from different direction (i.e. - top to bottom, bottom to top, left to right, etc.). You should notice that it selects different sectors when you cleave from different directions
Next we'll cleave a surface. Switch to surface mode and make sure you're in a top view. Select a surface and do as above; clicking and dragging and from different directions. Pretty easy.
Sector and Surface cleaving in action.