There are several types of backstabs. They all work on the principle that your knife hit the enemy in the 180 degree arch behind them. Ignoring the stabs that involve sapping, here are a few common terms I've heard and/or used. All these terms are relative to my experience. Some terms may have multiple names and is subject to interpretation:
(Note: the idea behind some of the stabs involves a more in depth look at backstabbing, server ticks and client/server registration. See this post and this youtube video by tf2 spy player Stabby Stabby)
Understab: The spy stands underneath a drop off point (like near the second blue spawn area in pl_badwater) and waits for players to jump/walk off the edge. As they are falling they pass over the head of the spy and expose their back for a stab.
Jumpstab (Leapstab or Dropstab): The spy jumps from a high, generally unseen, height and stabs the enemy. This is generally done as a surprise attack on high value targets. Think like jumping off the tower on plr_hightower onto the ground below to stab.
Topstab: This stab is generally done on engineers. If an engineer has built his sentry and dispenser in such a way that his back is to the wall but he still has access to his sentry and dispenser for repairs you need to stand on his head. Once on his head you back up as much as possible and stab him. This is generally done with a silent decloak with the saharan spy set or when the engineer is distracted and can't hear a decloak. Don't forget to sap!
Cornerstab: As an enemy comes around a corner they generally are already facing the way they want to be going (as opposed to getting to the corner, stopping, and turning 90 degrees). As a spy, you anticipate this and as they are just coming around the corner, move to their side quickly and stab before they can turn.
Stairstab: As an enemy is going up a flight of stairs, the spy on the higher ground jumps above the enemy and gets behind them and stabs. This can also be done on almost any vertical height displacement like from boxes, engineer buildings or even small rocks on the ground. The spy simply needs to be able to jump over the opponent.
Sidestab: This stab generally relies on the enemy buying into your disguise and not bumping into you. As an enemy passes by you going the opposite direction you are traveling, very quickly turn and stab. Works well on fast targets like scouts and medics.
Matador stab (Jukestab): Similar to the sidestab but works more on the notion that the enemy doesn't buy your disguise or you are not disguised and in combat. You approach the enemy, or they approach you until you get into melee range. Strafe one direction very briefly then switch sides strafing and stab. The first strafe is a fake out and forces the enemy to over extend their field of view and allows your second strafe to get to their open backside for the stab. I find this works well on heavy's shooting a minigun or enemies doing melee combat.
Strafe and Snap: This is a variation of the Matador stab. Instead of doing a quick first strafe to fake them out and the second strafe to do the stab you simply tighten up the timing and use the first strafe to get the backstab. Requires very fast timing. This was coined by the player Stabby Stabby.
Facestab: This is not an actual backstab technique but a phrase coined by the community to descrive backstabs that shouldn't have worked. For most of the stabs listed above the view from the enemy is that they were looking at you and then suddenly they were backstabbed. If they were looking at you how could you hit their back? Hence the word facestab. In reality, most facestabs are the difference in server/client registration and server ticks, which is explained in a link above. On the spies screen the back was shown and the stab looked fine. On the enemy's screen the spy was clearly in front of them and couldn't possibly be at their back.
Failstab: This is a stab that should have been a backstab from the spy's point of view but didn't register as a backstab and only as a regular melee strike. This is due to server/client registration and server ticks. Think of the failstab as the opposite of a facestab.
Butterknife: This means to just keep stabbing an enemy with the knife until they die from regular melee strikes. Generally this is done on low health enemies who would die from a regular melee strike (with health usually <40). It could also be an act of desperation from a spy attempting to stab an enemy where running away is not an option. This involves generally two things: 1) Swinging your knife wildly and hoping that you get an accidental matador stab and 2) Calling our your butterknife after it was successful. Example: "Dude, I totally butterknifed you!"
Most stabs and a few others are found here.