Daemati are faeries that have the ability to control and infiltrate another person's mind. They are able to read, influence, and even shatter someone's mind, depending on their training and power.
Daemati are very rare, as the trait appears as the Mother wills it, so the faeries that have it are strongly trained and are feared. There are enough scattered throughout the world that many, mostly those in positions of influence, extensively train against a daemati's skill set.
Depending on how much a daemati is trained, the power can vary from simply slipping into someone's mind, like Feyre Archeron did with Rhysand and Lucien Vanserra, with or without wanting to, to shattering someone's mind, effectively driving them crazy and leaving them a mindless zombie. Or they can make people their unwilling slaves, making them do whatever they want and the person would never know it.
If a daemati is able to get past someone's mind shields, they can see through that person's eyes, as Feyre does with Tarquin, or read their thoughts and alter them as they please.
When Feyre is rescued by Rhysand on the night of Calanmai, they make a small connection. Later, Rhysand uses his daemati powers to read Feyre's mind and deduce that she is lying about her name in Tamlin's residence when he discovers she was being glamoured by Tamlin.
When Rhysand saves Feyre from her wedding with Tamlin, he finds out that she got a piece of each High Lord's powers. He teaches her to create a mind shield to shut him, or any other daemati, out of her mind.
The first time she uses her powers is in the Spring Court when she accidentally enter in Lucien's mind. Then she does it again, in Velaris, with Rhysand's mind and for the first time since Amarantha she sees how she has changed. Her face is gaunt, her cheekbones are sharp, and her eyes are dull and smudged with purple beneath. Her lips are wan and her collarbones jut above the neckline of her sweater. She looks as if rage, grief, and despair have eaten her alive, as if she was again starved. Not for food, but for joy and life. She asks Rhysand if he is doing some kind of trick but he says no. They don't know how she was able to get through his shield. He hypothesizes that perhaps since her daemati power comes from him, if his own shield mistook her for him and let her slip past.
She also uses her powers to enter Tarquin's mind while in the Summer Court when she is trying to steal the Book of Breathings.
When Feyre is back in the Spring Court, Rhysand's lessons prove to be useful, because she encounters two daemati: Brannagh and Dagdan. She uses her new skills to protect Lucien and Tamlin from them. Because she was so focused on protecting them, she didn't realize that Brannagh and Dagdan were slowly weakening her powers by putting Faebane in the food and drink.
Feyre uses her daemati powers when she sees Lucien and Ianthe. With her powers she forces Ianthe into shattering her own hand with the rock.