Manipulating Brushes

You'll notice that when you make a spike, cylinder, or wedge they are always orientated in the same direction. You can use rotate in the appropriate window will make the object face the desired direction. In [Tools => Options] 2D views tab, you can click "Reorient primitives on creation in the active 2D window". That just orients the block to the last 2D window you clicked in before you pressed [Enter] to create your object. You will want to play around with this a bit to get used to it.

Clip Left

Clip Right

Split

Now, you can do three different things.
There are three different clip modes, Clip Left, Clip Right, and Split. The current mode can be changed by repeatedly clicking on the Clip Plane button. The red portions of the brush is the part that will be discarded. To perform the clip, all you need to do is press [Enter], after making sure it is set how you want.

It is possible to clip across any number of brushes, as long as they are all selected beforehand.
It's also possible to move the clip line around by holding Ctrl and click-dragging it, so both clip points will move at once.
To start a new clip plane, hold Shift and click drag a new line.

The Clip Tool is being considered by id software as a replacement weapon for the Lightning gun for Quake2
:)

Vertex Manipulation Mode

6 - Invalid Brush (this is bad)

Vertex manipulation is quite simple to use. Select a brush (1), and press the Vertex Manipulation button. The brush will turn translucent. The vertices will be marked in white, and the edges will be marked in yellow, currently selected points will be marked in red (2). You can move any of the points in either the 3D or 2D window by clicking and dragging. Click the select mode button to switch back to normal select mode.

Vertex Mode Face Splitting
With the registered version, you can also split the face of a brush. Select two edges (2), and press Ctrl-F. This will turn the edges into vertices, and create new edges, splitting the face (3). You can select the new edge and drag it outward (4). Click X to switch back to normal select mode. This is a valid brush (5).

After you finish using vertex manipulation, you must switch back to normal select mode, or the manipulations will not appear in your level if you compile it right away. 

One must be careful with vertex manipulation though. If the edge was moved in the opposite direction in this example (6), it would form an invalid brush and would cause errors. These errors can often be found by using [map => check for problems], so it is important to use this after using vertex manipulation. It is better, however, to learn how to not make invalid brushes. Simply put, if a straight line can pass through more than two faces of a brush, it is invalid. You may need to use two separate brushes to make your intended shape.

You can split the faces of a brush any number of times, however the more splits you add, the higher the likelihood that you may create an invalid brush.

Carve

Carving is simply removes or subtracts the volume of the selected brush from any brush it is within. The brushes which are carved are often split into fragtments or splinters to allow the remaining brushes to be valid shapes.

Something that some level designers who use carving will do is to copy the the brushes to be carved into a new map, and do their carving there. Then copy the finished product back into their good map. I'm not aware how much this helps, because I don't use carving any more. Most brush actions and shapes can usually be accomplished without the use of carving.

There is an excellent section covering carving carefully in the Forge's Tips and Tricks Section.