Let's discuss how to play Veigar on this current patch, 13.20.
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Veigar is a hard scaling burst mage, who also happens to have one of the most valuable teamfight CC abilities in the game, Event Horizon. Also known as "cage."

Thanks to both his champion passive, and his Q passive, Veigar can collect an indefinite amount of phenomonal evil stacks, which in effect, gives him indefinite bonus AP throughout a game.

Since his ult is a single-target, point and click ability, with an execution mechanic, Veigar's infinite scaling gives him the potential to single-handedly take down just about any target during the late game.

With this, it's easy to classify Veigar as a kind of "burst assasin mage," BUT, to do that, would completely discount the utility and team fight power of Event Horizon. Thanks to Event Horizon, there is a strong arugment to be made that Veigar can be classified as a true "Control Mage," with viable support builds that can max E and build for survivability.

Veigar does have a few glaring flaws that can certainly be exploited on this patch.

First of all, Veigar's early game is very weak, and you may find yourself feeling like you are running "behind" for a while, which can be exasperated in matchups against champs that out-range you, or champs that can out-roam you in the early game.

Second, Veigar is highly immobile. If you use cage to harass someone, you better be on high alert for enemy ganks, as you have essentially no escape. Even with cage, highly mobile enemies can sometimes be very tricky to reliably lock down.

Third, although you can indefinitely scale your AP, there are still enemies that will outscale you on the current patch. Aurelion Sol immediately comes to mind, and Riot seems to agree, as he's already slated for a nerf on the next patch. Some champions with kits that do more than Veigar, or kits that innately scale, can be a problem late game, even for Veigar.

Veigar's passive is called Phenomenal Evil Power. Every time you hit an enemy champion with an ability or score a takedown, you gain permanent ability power.

Notably, this passive is proc'd by all 4 of your abilities, and even landing a cage on an enemy champion will give you AP. This is part of how even Support Veigar can obtain scaling in lane.

Veigar's Q is called Baleful Strike, and is a linear skill shot, which damages the first two enemies hit. Kills with this ability grant 1 AP, which is tripled against LARGE minions and monsters.

If you're looking to scale hard, last-hitting with Veigar's Q is incredibly crucial. It cannot be stated enough. In fact, if you take only one Veigar lesson from this whole video, it should be this:
Farm. Your. Q.
Farm your Q. Farm your Q. Farm your Q.

You're a champ build for hard scaling, so don't be weak.

Since leveling Q lowers its cooldown, and increases damage, this is generally the ability you'll want to max first, as it will make farming much easier. You'll also find, as you scale, that this ability can provide reliable damage in team fights.

Veigar's W is an AoE damage ability called "Dark Matter." After a brief delay, dark matter falls from the sky, damaging enemies.

This ability, while hard to land, can be very useful at times to zone enemies. It's also a perfect follow up to hard CC, such as Event Horizon.

Event Horizon, as we've discussed, is your E. It creates a cage that stuns enemies that touch its edges.

As hinted at earlier, this is essentially your only self-peel, so always be sure to weigh the risk and cost-benefit of using it during the laning phase, or when you are far from safety.

This ability has seen numerous hard nerfs over the years, but arguably still remains one of the best non-ultimate CC abilities in the game.

Veigar's ultimate is Primordial Burst. This is an execution finisher move, and the lower the enemy's health, the more damage it deals.

Being a point and click nuke on a single target, it's similar to the ability Flare from Final Fantasy, which is quite fitting for a black mage.

Currently, Veigar can be played comfortably in either mid or bot. Support Veigar is also viable, but is not currently a very popular or "meta" pick, and it can have a bit of a different play style from mid or bot.

Essentially, Veigar has one "mini combo" that you will benefit from being familiar with, but it won't require drilling in the Practice Tool.

The important "mini-combo" to be aware of as Veigar is Q > Flash.

By starting the Q animation before flashing, you can buffer the animation and catch enemies unaware. This will give them much less response time than if you were to flash and then Q instead.

Point your mouse where you want your Q to go, and then start the combo, pushing Q first, and then flashing.

This mini-combo is not only great on Veigar, but translates to many other champions as well, such as the Ahri charm.

When chasing an already wounded target, you can often follow this up with an immediate ult, to secure a kill.

Aside from this, Veigar doesn't have many classic "combos," as it is really just about spell rotations.

I will note here that cage placement is an incredibly crucial part of playing Veigar, as a good stun will allow you to follow up with the rest of your abilities.

If running Everfrost, be sure to pair it with Event Horizon for chain CC, so you can extend your trades and achieve more damage.

And one note on your ult, Primordial Burst: As already mentioned, this is essentially an execution ability, as it does more damage based on missing health. For this reason, you should always try to hold Primordial Burst for the end of your trades, when you know you can finish an enemy, or for finishing specific priority targets on the enemy team.