Illaoi in Set 10
Illaoi is a new Legendary Tier 5 Unit coming with Set 10. She has a unique trait called Illbeats, that spawn two Tentacles if you field her. You can freely move these Tentacles and they will interact with her ability cast.
If you've played Set 4, then you should be familiar with this type of unit, since Warlord Azir had a very similar mechanic. Receiving two placeable units that are unable to move opens up a surprisingly high amount of possiblities to give yourself an advantage.

Counter: Karthus
When you are playing Set 10 you will encounter Kartus a lot. He is a Tier 4 AP Carry with powerful traits and an ability that can access the backline. Karthus will target the five lowest HP units and deals a high amount of magic damage after a short channel.
If you're facing a 6 Executioner Karthus then every non tank unit that is focused will likely die in a single cast. To give yourself a fighting chance you need to field Illaoi against him and make use of her Tentacles. Both of the Tentacles have quite low base HP and are therefore almost always the main target for Karthus. Running a single Illaoi will make Karthus waste almost half of his ability giving you more time to turn the fight.

Counter: Akali
Another unit you will face a lot is KDA Akali. She is the cloest we have to an Assassin in Set 10, since she will dash to the farthest enemy from her position. This makes cornering units very dangerous against Executioner Setups, since she can bring your corner unit low enough to get prioritized by an upcoming Karthus Ult as well.
This is where Illaoi proves herself yet again as a great counter. By putting the Tentacles in each corner, she will end up wasting her cast and not build up any repeat dashes either.

Maximizing Combat Augments
Since you are free to move Illaoi Tentacles in the Planning Phase, they are capable of interacting with a lot of combat Augments.
Both Tentacles can be used to trigger the bonus Attack Speed and Armor from Best Friends.
Both Tentacles can be used to create a larger shield with Keepers.
The first two Tentacles can distribute and receive the HP Shield and Attack Speed from Inspiring Epitaph.

Final Tips
Something that is also neglected a lot is that Illaoi shares her Armor and Magic Resistance with her Tentacles. This means there are a lot of ways to "double dip" defensive stats. For example if you run 2 Sentinel, then both the Tentacles and Illaoi will receive +16 Armor/MR, but then the Tentacles will receive an additional +16 since they take Illaoi's bonus stats as well.