There are four basic tools used to transform brushes, of which only two are frequently used, these are the Default or QE activated by clicking , the rotate tool activated by clicking or by pressing R, the scale tool activated by clicking and the transform (move) tool activated by clicking or pressing W.
The QE Tool 
The QE tool (lets call it that) is the tool you will use the most when transforming brushes, it can move, create and scale brushes (not in the same way as the scale tool though). To move a brush with the QE tool, simply click within the brush (after selecting it) and drag the mouse.
Creating brushes is also as simple as dragging the mouse over the grid (while holding down the left mouse button). The newly created brush will then appear where the last selection was. Now if this is hard to understand, the brush can only be created in two dimensions, which are different depending on what view you are in, the third dimension is then taken from the previous selection. If you still don't get it, try it out.
The final function of the QE tool is to scale brushes, if you have ever used a 3D modelling software such as 3Ds Max or Blender, you will realize this is similar to the Extrude tool of those applications. You can scale a brush by clicking outside the brush to scale (after selecting it) and dragging the mouse while holding down the left mouse button. This will then extrude the brush where you drag. You will use this a lot.
The Rotate tool 
The rotate tool, also called free rotate is used to (guess what) rotate brushes freely. However this usually distorts the texture so it's not really recommended to be used on stuff where the texture alignment matters. However it's good to rotate things like rocks where it doesn't matter. For things where it matters, use the rotate and flip tools instead. They will make sure the textures stay aligned.
To use the rotate tool, drag the lines to rotate along one axis only and drag between the lines to rotate the selected brush freely.
Note: This tool is very useful for rotating models since the textures don't get disorted there
The Scale tool 
The scale tool is used to (getting used to it by now eh) scale brushes. However, like the rotate tool, this tool will also mess up the texture alignment. Usually, it's better to use the QE tool to scale brushes.
To use the scale tool, drag the colored lines to scale along one axis and he yellow box to scale along all axis.
Note: This tool is very useful for scaling models since the textures don't get disorted there
The Transform tool 
The transform tool should be called move tool, since that is what it does, it moves stuff, entities, brushes, models. When the tool is activated, three big arrows, one for each axis will appear in the 3D view, and 2 big arrows will appear in the grid (2D) view.
Dragging these arrows will move the object back and forth along the respective axis, dragging the cyan box in the middle will allow you to move the selected object in any direction.