Editor toolbar's tools, aka toolbox

Global Edit mode

Ardour has a global edit mode selector at the left of the Editing toolbar, which affect how regions are moved or copied:

Slide
Regions move freely. Ardour creates overlaps when necessary.
Ripple
Editing affects the regions to the "right" of the edit (see below).
Lock
No region motion is permitted (except for "nudge").

Ripple Edit mode provides the following conveniences:

If Snap To Grid is enabled, then regions can only move so that they align with locations determined by the current snap settings (beats, or seconds, or other region boundaries, etc). See Snap To the Grid for details.

The Smart switch

The Smart Mode button to the left of the mouse mode buttons modifies the Grab Mode. When enabled, the mouse behaves as if it is in "Range Mode" in the upper half of a region, and in "Grab Mode" in the lower half. This allows avoiding constant switching between these two modes.

Mouse Modes

Grab Mode
The Grab Mode is used for selecting, moving, deleting and copying objects. When in object mode, the mouse pointer appears as a hand whenever it is over the track canvas or the rulers. The mouse can now be used to select and perform operations on objects such as regions, markers etc. This is the most common mode to work in, as it allows you to select and move regions, as well as modify automation points on the automation tracks.
Range Mode
When in Range Mode, the mouse pointer appears as a vertical line whenever it is over the track canvas or the rulers. The mouse will now be able to select a point or range of time. Time ranges can be selected over one or several tracks, depending on the selection of your tracks.
If none of your tracks are selected, the Range Tool will operate on all the session track visualized in the Editor.
If you want to edit only particular tracks, select them before you apply the range tool.
Cut Tool Mode
When in Cut Tool Mode, the mouse pointer appears as a pair of scissors whenever it is over the track canvas or the rulers. This tools allows to cut any region into 2 regions at the mouse cursor, regardless of the Edit Point.
If one or more track(s) is selected, then all the regions on these tracks will be split at the mouse cursor position.
If no track is selected, then only the region hovered by the mouse cursor will be split.
Stretch Mode
When in time fx mode, the mouse pointer appears as a distinctive expanding square symbol whenever it is over the track canvas or the rulers. This mode is used to resize regions using a timestretch algorithm. Click on an edge of a region of audio and drag it one way or the other to stretch or shrink the region.
Audition Tool
Clicking a region using the audition tool will play this region to the control room outputs.
You can also scrub with this tool by clicking and dragging in the direction you wish to listen. The amount you drag in one direction or the other will determine the playback speed.
Draw Tool
When in Draw Tool mode, the mouse pointer will change to a pencil. You can then click within an audio region to change the gain envelope for that region. This curve is separate from fader automation for individual tracks. It will remain locked to the region's time, so if the region is moved, the region gain envelope is moved along with it.
The draw tool works on automation too, allowing the creation and modification of control points on the automation curves.
Last, it is used on a MIDI region to edit the notes.
Internal/Region Edit Mode
When in Internal Edit mode, the mouse pointer will change to cross-hairs. This tool acts on gegion gain and automation as the Draw tool.
On a MIDI region, it allows to lasso-select multiple notes at a time.