When you're ready to edit your objects on a component level, you'll want to enter Edit Mode. Do this by selecting your object and hitting <Tab>. Try this with the default Cube. Your Cube will then be orange with all vertices selected (Figure 1). It is in vertex mode by default.
Figure 1: Cube in Edit Mode will all vertices selected
Before we explore edges and faces, look to the top of the interface, to the right of the "Blender Render" drop-down menu (Figure 2). This area shows how many total components there are in the object and how many are selected. For example: it says "Ve:8-8 | Ed:12-12 | Fa:6-6 | Cube" for your selected Cube. "Ve" stands for vertices. The first number is how many are selected, and the second number is the total in the object. If you hit "A" to deselect all, each first number will then become zero.
Figure 2: Selected Components
Now, let's look at the other components. Locate the icons at the bottom of the view, to the right of the "Global" drop-down menu (Figure 3). From left to right, the icons are "Vertex select mode," "Edge select mode" and "Face select mode." Also, note the next icon to the right: "Limit selection to visible" (Figure 4). This is like Backface Culling and is good for avoiding selecting components on the back side if you do not wish to.
Figure 4: Limit selection to visible
Switch to Edge select mode by clicking its icon; the vertices will disappear. Then click on Face select mode by clicking its icon to see the "tick marks" appear on the center of each face. You can also have multiple modes active at one time: hold Shift and click another component mode icon.
Now that you know how to access the icons, memorize this hotkey: CTRL + <Tab>. This will take you to a quick menu "Mesh Select Mode" to select mesh components (Figure 5). Make sure you're in Edit Mode before you use this hotkey, or you will enter Weight Paint Mode.
Figure 5: Mesh Select Mode menu
With any vertices, edges or faces selected, you can then use the transform gizmo or hotkeys to position, rotate or scale the components, just like you would do in Object Mode.
The selection rules apply in Edit Mode as well. Unless you've changed your Preferences, RMB will select components, Shift + RMB will add or subtract components to the selection. "A" toggles select all/deselect all. "B" activates the Border Select Tool (marquee select). "C" activates the Circle Select Tool (brush select). Your first and last selection (active) will be white if vertices or edges and gray, if faces.
Try adding some 2D views and hit "Z" to go into wireframe mode. Using the Border Select Tool to select a line of vertices, etc (Figure 6). This will have the same effect as other 3D programs; you'll use it to marquee select many components quickly.
Figure 6: Border Select vertices
There are some other selection methods as well. Go into Object Mode and delete the Cube. Don't do this in Edit Mode, or you will be creating a new mesh within the Cube object. After the Cube is deleted, Add a UV Sphere to your scene. Tab into Edit Mode and Limit Selection to Visible. Another suggestion is to hide the transform gizmo, as it tends to get in the way when you're selecting components. Deselect its icon or just hit "CTRL + <Spacebar>." Deselect all vertices before we begin.
Certain mesh objects allow you to select "loops." Let's select an edge loop. You can actually do this in either Edge mode or Vertex mode but to avoid confusion, change to Edge mode. Using Alt + RMB, select an edge. If you selected a horizontal edge, a horizontal edge loop is selected and vice-versa (Figure 7). Rotate your view around to verify this.
Figure 7: Edge Loop selection
Go into Face mode and do the same thing. Face loops are selected by using the edge in between the face direction you want. For example, Alt + RMB-select a vertical edge between two faces. A horizontal face loop is selected all the way around the sphere. Deselect all, then Alt + RMB-select a horizontal edge between two faces. This will select a vertical face loop, but vertical face loops on a sphere only go half-way around (Figure 8).
Figure 8: Face Loop selection
You may also invert a selection by hitting CTRL + I. Try this to see that the vertical face loop is now the only deselected part of the sphere.
Let's say you now wanted to delete the selected faces. Again, use "X" or <Delete>, just like normal. When the Delete menu pops up, choose "Faces" to delete the selected faces (Figure 9).
Figure 9: Delete Faces
Tab back into Object Mode to see your new object. Hit "G" and move the object over a bit, then Tab back into Edit Mode and Undo with CTRL + Z. Blender keeps track of the undo, even after you've exited Edit Mode. Even though you moved the object, you can still undo changes in Edit Mode.
One last area we'll touch on is the Center Point, or the "Origin" of the object. The object's origin is its true pivot point. With your object still in Edit Mode, select one face and hit "N" to bring up the Properties panel. Look at the Transform area, under "Median" (Figure 10). These coordinates describe the face center's location in relation to the object's origin. It happens to coincide with the Global origin right now. If you switch the button below from Local to Global, no change will appear. However, if you Tab back into Object Mode, move the object over a bit, Tab back into Edit Mode and look at Global, it will have changed because the object has moved in relation to the Global origin. But, if you switch back to the Local button, it will still be the same, because the mesh geometry has not changed in relation to its own origin.
Figure 10: Creating a Top View
An object's mesh data will also keep track of how the components are orientated in relation to the object origin. If you move, rotate or scale your object while in Edit Mode, you have changed the object on its mesh level. This will, in turn, affect the object origin. Again, think of the object origin as its pivot point. The pivot point is in the center of an object, by default. If you go into Edit Mode, select all vertices and move them, your pivot point might lie outside the mesh. If this happens, hit CTRL + Alt + Shift + C to bring up the "Set Origin" menu. Then you can either choose "Origin to Geometry" to put the pivot point back in the middle of the mesh or "Geometry to Origin" to pull the geometry around the location of the pivot point.