Summary
For every character, a Launcher is performed just by hitting the Special Button while on the ground. It does not matter what direction you are holding on the controller, as long as you are on the ground when you hit Special, you'll perform the Launcher.
Some characters have extra moves that also count as Launchers. Examples of this are Doctor Doom's Doom Kick (Toward + Hard Attack) and Chun Li's Hoyokusen Hyper Combo.
Once a Launcher connects, just press any of the Up directions on the controller to Super Jump automatically. There is no need to tap Down on the controller as you would when performing a Super Jump normally.
Aerial Rave
Launchers will always knock the enemy high up into the air a fixed distance so that your follow-up Super Jump can be used to land an Air Combo. The height the opponent is launched up is dependent on each character's Launcher. The Launchers work regardless if you hit the opponent on the ground or out of the air. However, keep in mind that because the distance they are launched is fixed, hitting them out of the air will put them much higher than if you hit them off the ground.
Once launched, you actually have a fairly large window on when you can perform your Super Jump into the air. You can hold Up even before the Launcher connects to Super Jump at the very first possible moment. Or you can let the Launcher connect and wait a brief second and then Super Jump after them with a tap of Up.
Strategy Corner
There is actually a practical reason for delaying your Super Jump at times. Air Combos in Marvel Vs. Capcom 3 are not as automatic as they were in MvC2. In MvC2, everybody pretty much could Super Jump almost immediately after a 
and land a full Air Combo on the opponent relatively easily. Not so in Marvel Vs. Capcom 3. Some Air Combos are better performed by waiting a split second before you Super Jump. For example, with Spencer, oftentimes if you Super Jump the 
you connect with your Launcher and try to land a Medium Attack into Medium Attack into Hard Attack into Wire Grapple, Spencer tends to fly past the opponent after the two Medium Attacks and the Hard Attack will whiff. However, if you hit the opponent with your Launcher and wait just a split second before going into the air, landing that Air Combo becomes much easier because they are now at a proper height at which the Air Combo will connect without having Spencer pass the opponent up.
However, if you the opponent out of the air, they already started at a slightly higher height, so performing an immediate Super Jump 
with Spencer might actually be preferable! Some characters can even do this on purpose, such as Storm. Hitting the opponent with her Crouching Hard Attack, which pops the enemy into the air, and then chaining into her Launcher is preferable to just launching the opponent directly. It makes connecting the hits in her Air Combo much easier when you the enemy from higher up. Obviously, you'll have to experiment with all your characters on the best timing to land you Air Combos fully.
Launchers also set up special properties with Air Combos. Air Combo Finishers are only possible after Launchers, as are Aerial Exchanges. So they definitely put the opponent into a special "Launched State" that sets them up for specific results.
Blocked or Whiffed Launchers
On block or whiff, however, Launchers cannot be canceled into a Super Jump. So if you start performing a Chain Combo on the opponent and they block the attacks, do not cancel the last Normal Move into a Launcher. If you do and it is blocked and it puts you in a delay, your opponent can counter you and there really isn't anything you can do about it (save for calling an Assist before the Launcher connects to protect you after the opponent blocks your Launcher).
Launchers Are Special Moves
Launchers, though they cause no Chip Damage, are actually categorized as Special Moves in Marvel Vs. Capcom 3. If you noticed in the Magic Series section of this guide, there was no discussion on Launchers and where they fit in to the Magic Series. That's because canceling a Normal Move into a Launcher isn't performed by Chain Combo, it's performed by Buffering, just like canceling a Normal Move into a Special Move.
The way you can tell this is because there are many Normal Moves in the game that cannot be Chained into another Normal Attack but can be Buffered into a Special Move... and every Normal Move that has this property can also be canceled into a Launcher. And moves that cannot be Buffered into a Special Move also never have the ability to be canceled into a Launcher.
More evidence is that you cannot actually perform a Cross-Over Assist during a Launcher, you can only do it before the Launcher itself is performed. Also, Special Moves and Launchers fall into the same Combo Damage Reduction category. All these factors point to the notion that Launchers are considered Special Moves and not Normal Attacks.
And a very key factor that points to Launchers being Special Moves is that they cannot be canceled into Special Moves themselves! So if you ever perform a Chain Combo on the opponent and end it by Buffering into a Launcher, you'll notice that you cannot save yourself by canceling the Launcher into a safe Special Move! Whereas before in MvC2 you could do an attack with, say, Captain America and go for a Launcher at the end of the Chain Combo, if the opponent blocked the Chain Combo you could always cancel the Launcher into a Shield Slash to save yourself from delay. This is not the case in Marvel Vs. Capcom 3.
Jump Cancels
If the Launcher connects on the opponent and Launches them into the sky, you can Super Jump Cancel the Launcher into a Special Move by canceling the Pre-Jump frames of the Super Jump into the Special Move. The easiest way to do this is by performing a "Tiger Knee" motion on the Special Move. For example, if you connect with Taskmaster's Launcher and want to follow it up with a Hard Attack version of the Air Master, where he shoots the arrows straight up into the air, you can do this by connecting with the Launcher and then pressing Down, Down/Toward, Toward, Up/Toward plus Hard Attack. The Up/Forward motion will cause you to start the Jump but since you pressed Hard Attack after a sequence of Down, Down/Toward, Toward, the Pre-Jump frames get canceled into the Air Master. But remember, you can't do this on block because if a Launcher is blocked, you can't cancel it into a Super Jump.
No Hit Stun Deterioration
Finally, Launchers are immune to Hit Stun Deterioration (see Hit Stun Deterioration section). No matter how far along into a combo you are, if you manage to hit the opponent with a Launcher, they will reel the same length of time and get launched into the air the same height no matter how much Hit Stun Deterioration has kicked in.