How does Animation Canceling work? Is it a valid technique?

Animation canceling is pretty important in dota 2. It serves several purposes such as last hitting better,optimizing the time an enemy is stun by one of your spell or the most important thing in my opinion : Orb Walking. Orb walking consist roughly in moving toward your target as soon as the damage is dealt (either you are melee or range doesn't matter) : thus animation canceling in order to reduce the gap between you and the target. By managing this you will avoid enemy running away because your hero has to stop moving when he attacks: See in this video how drow keeps up with vengeful spirits.

You can animation cancel in 3 situations :

• After an attack : Each hero in the game has a different timing for animation canceling. You have to learn for your character when the attack animation begins, and when the damage is actually dealt to your target. Its easier to tell for ranged heros as you can cancel the animation when the projectile goes.
• After a spell : You have here to learn when the spell actually starts (or when the damage is done) before animation canceling. One of the biggest example is crystal maiden's frostbye. You can animation cancel when you see that the ennemy stopped mooving, if you waited the whole animation you would have about 1sec less to attack him while frozen.
• During a spell that has an obvious and quite long animation (Leshrac's Split Earth, Eartshaker's fissure, Magnus' Reverse Polarity etc.). By pressing the "Stop" key during the animation, you can fake the cast of a spell thus baiting a defensive response from your opponents. The spell won't be actually casted saving you the mana and the cooldown but it can bait a blink, Puck's phase shift or if you are lucky, a Black King Bar use.

You can animation cancel :

• After an attack by mooving (easiest way to achieve it)
• After a spell by mooving or pressing the "Stop" key (be careful to not cancel your spell, it's more important that your spell hit than animation canceling). In some circumpstances casting another spell will animation cancel to. With Crystal Maiden's Frosbyte, you'll be able to cast Frost Nova as soon as the ennemy stopped mooving thus animtion canceling the end of Frostbyte cast.
• During a spell animation in order to bait : only by pressing the "Stop" key. If you try to animation cancel a Fissure by mooving you will cast your Fissure first.

It is an excellent technique however it requires a lot of practice. You dont have to think about it when you play but do it automatically. Train yourself with the character you are the most confident with, only practice will really teach you this mechanic of the game.

DotaCinema made a pretty good video that shows the concept more precisely :

Animation cancelling is technique used when you try to get faster attacks. Let me try to explain this by taking Drow Ranger as an example.
Before skilling her ultimate (Marksmanship), her attack speed is too slow. She takes some time (by 'some time' i mean noticeable time) to lift her bow, then fire the arrow and relax her bow. Now what animation cancelling aims to get done is to eliminate the time spent to 'relax her bow' , ie, the time spent (read: wasted) after the arrow has been fired and before she raises her bow for the next attack.
Well, Animation cancelling is useful only in the early phases of the game when the hero's attack speed is low, and you can reduce the 'wasted' time. Once the attack speed crosses a limit, there is no point in animation cancelling for faster attacks.
How is this achieved?. As soon as the shot/attack is fired, move to spot very close to the hero's current position and give an attack command (A->left-click or right click the creep). The time spent on moving to the new spot and giving the attack command has to be lesser than the 'wasted' time.
Time to move + time to give attack command < Recoil time of the previous attack. Animation cancelling is useful only if the above equation holds. If you move to a spot quite some units away, then the time spent there will be way more than the recoil time and the whole purpose of animation cancelling is undone.

PS: If you are a safe lane carry, then it is advisable to just take the last shots, since continuously attacking with/without animation cancelling will push the lane and reduce you farm capability.
Hope this helps!

Edit: This was meant to explain only the last-hit animation cancelling. The answer procided by @WizLiz covers the different types involved.