According to this question (better citation needed), Yoda said to Luke:
The Force is strong with your father, you, and your sister.
Did Leia ever use The Force?
Within the movies - there's a couple of points where there seems to be an affinity between Leia and Luke. Whether this is because of Leia's latent talent, or Luke's own force ability, is up for debate.
She gets Lando to turn the ship around to rescue Luke from the bottom of Cloud City, because she "hears" him call out to her
She "knows" that Luke got off the Death Star II before it exploded
In the Expanded Universe, however, there are many examples of Leia developing her force ability. While these are not the same canonicity as the movies themselves, EU canon is still held to be strong wherever a movie doesn't outright contradict.
There's at least two examples of Leia actively using the Force in the official novelisations for 'A New Hope' and 'Return of the Jedi', during her interrogation by Vader and once again on Cloud City.
“Where did you send them? Where is the Rebel base?” Without warning, the fire in Leia’s body was blown out, replaced by an icy grip. Fear wound its way through her, snaking down into her stomach. The feeling was so different from the light-headed dream of the serum. There was…power behind each of his words. They nudged at her. They prodded as sharply as any knife. So Leia did the only thing she could—she pulled back. Physically, toward the wall. Mentally, to a place the voice couldn’t find her. An unfamiliar warmth wrapped around her, a protective blanket that didn’t let any of the darkness nudge through.
The freezing pressure on her mind was thrown back. The rumbling voice made a sound of surprise and was quiet for a long moment. “If you do not tell me where to find the Rebels, lives will be lost! All of the Rebel deaths will be on your head!”
Hey,” he jostled, “I’ll bet Luke got off that thing before it blew.” She nodded. “He did. I can feel it.” Her brother’s living presence touched her, through the Force. She reached out to answer the touch, to reassure Luke she was all right. Everything was all right.
In The Force Awakens, Leia uses the Force passively on two other occasions.
She senses the deaths in the Hosnian System
Leaning against a console for support, she steadied herself. “A great disturbance—in the Force. Deaths and passings. Too much death, too many passings.” Straightening, her expression grim, she walked over to confront the wiry, slight Admiral Statura. Despite his experience in battle, he was left as shaken by the revelation as anyone else in the room. What had just happened could scarcely be comprehended.
She senses the death of her (ex?)Husband, Han Solo
"On another world far, far away, a woman felt a shudder in the Force that lanced through her like a knife. She slumped into a seat, her head lowering, and started to cry."
In The Last Jedi, she uses the force to survive
This is pretty much the most overt use of Force power exerted by her in any canon media.
“Call it whatever you want.” With that, Leia lifted her blaster, losing her sights on Rinnrivin’s guard—
—and targeting the central strut of the tunnel support directly overhead. One bolt held the entire thing together. That bolt was no larger than a child’s fist. At this range, in semi-darkness, perhaps one shot in a thousand might be capable of destroying that bolt.
But Leia made the shot.
In the last Jedi
Leia manages to rescue herself from deep space using the force.
Which is probably the most obvious of her force usage in any of the films
In some of the novels, Leia eventually trains as a Jedi under Luke. Much to Han's chagrin.
See: Truce at Bakura, Courtship of Princess Leia.
Also, in Empire Strikes Back, Leia senses Luke's presence. Whether this is purely Luke, or synergy between the both of them, is a matter of conjecture.
And, in Ep VIII, she uses the force in an undeniable manner.
Leia is force - sensitive and although, she never trained to become a Jedi, she has used the Force numerous times.
In the canon Star Wars book, Aftermath: Life Debt, Leia senses her child is a boy as well as the dark side:
"...Her thoughts reach for her wayward brother like a living thing, like branches seeking he sun. I need you here. I need your help. Luke sometimes had a farm boy's naïveté, yes, but right now she feels she could use a little of that. Her mind is a tangle of thoughts. The complexities of politics, the love of (and anger over) Han, the loss of Luke, and above all else the ever persistent worry about the life she carries--- Her skin tingles. Her mind feels suddenly unmoved from the rest of her. Leia feels dizzy enough to fall over. Oh. Oh, my! There! There it is. Washing over her and through her---and awareness unlike any she's ever felt before. A pulsing glow, flickering and strong. It's not the plant. It's not Luke. It's not even Han. It's her child. This isn't just a mother's recognition of the life inside---that she already knows....This is something separate from her. It isn't a physical feeling. It is all around her.....As such, she is suddenly aware of her child's mind and spirit: She senses pluck and wit and steel blood and a keen mind and by the blood of Alderaan is this one going to be a fighter! Wait. He? It's a boy. It's a boy. Her hands fly in front of her mouth as she both laughs and cries at the same time. This, she thinks, is the light side that Luke always goes on about---the promise of light, the promise of a new life...and then, the black edging of the dark side encircles her bliss like a noose. Because what rides swift on the heels of hope but fear--a fear that stretches out far and wide like a growing shadow....."
Through several canon comic books, Leia is seen fighting with a lightsaber and She also sees Padme's protrait turn and stare at her. She also senses Maul's presence in the palace of Theed from where he stood years ago.
In another canon Star Wars book, Bloodline, there are other examples of Leia's force-sensitivity:
"...Leia mentally cataloged that name for later. Maybe her curiosity had only seized upon a random group of people, frequent gamblers who would naturally visit a world like this often, calmer than most. Her suspicion of them was only a hunch, but her hunches often proved to be right. Luke said those strong instincts of hers served as proof of the Force, evidence that it was working through her all the time..."
In the films, various scenes showed Leia's force-sensitivity:
Leia doesn't give away where the plans for the Death Star are located, despite interrogations and pain (Episode IV)
She knew the Millennium Falcon was being tracked (Episode IV)
Leia senses something is wrong in Cloud City and is distrustful of Lando (Episode V):
LEIA: Something's wrong here! No one has seen or knows anything about Threepio. He's been gone too long to have gotten lost. Han takes Leia by the shoulders and gently kisses her forehead
HAN: Relax. I'll talk to Lando and see what I can find out.
LEIA: I don't trust Lando
Leia hears Luke's SOS call through the Force (Episode V)
Leia remembers her mother (presumably through the Force) (Episode VI)
Leia always knew Luke was her brother (Episode VI):
LUKE: Yes, it's you, Leia.
LEIA: I know...somehow, I've always known
Leia senses Luke left the Death Star and is alright (Episode VI)
In The Force Awakens novelization, Leia senses the deaths on Hosnian Prime as the system is destroyed
She senses some light in her son, Kylo Ren (Episode VII)
She senses Han's death (Episode VII)
In The Last Jedi, Leia's force powers are expanded upon and we see her:
11. save herself by using the force to pull herself back to the command ship after the bridge she was in was exploded by the First Order. She senses Luke when he opens himself up to the Force again on Ahch-To while in her coma and later admits that she knows that her son is gone. Finally, Leia senses Luke's death along with Rey at the end of The Last Jedi
There may be more examples, however, these are the main ones that I can remember. Leia is force - sensitive, however it is more subtle than Luke's, only because she chose a different path than her brother's. J.J. Abrams has stated that they are leaving the door open for Leia's character, saying it is "never too late" for the character to pursue a Jedi path.
In the canon comic book Shattered Empire, Leia goes to Naboo, more specifically Theed Palace. While there, she senses the Force remnant of Darth Maul from decades ago, when he was there in the corridor she stood in.
In the 2017 novel by Claudia Gray Journey to Star Wars: The Last Jedi Leia, Princess of Alderaan, Leia makes use of the Force, although it was more "accidental" than anything as she has no idea of her origins at this time, and unaware of her Force potential.
Leia uses the Force to save herself and her classmate from falling into a ravine, both by using the Force to calm herself, and to be aware of her surrounding, and the novel also hints that she uses the Force to augment her own strength to help them swing to safety when hanging in the ravine.
Some interesting excerpts from the book that show the influence of the Force in this part
- As soon as she said that, Leia felt... centered, in a way she hadn't felt before. ...
She shut her eyes, breathed in again. It seemed to her she could feel the shape and dimension of the rock around them, that she knew the location of every spar and outcropping as automatically as if they were a part of her.
Leia leapt at the exact moment Kier did, her limbs flooded with strength beyond what she'd thought she'd possessed, They swung around the outcropping and easily reached the other side...
Keir: "That was one hell of a jump," he finally said. "How did you do that?".
Leia: "I don't know." Already that odd spell that had fallen over her was broken.The strange vital energy that had so briefly sung to her had gone quiet again.
In the force awakens trailer Luke clearly states:
"The force is strong in my family. My father had it. My sister had it. I have it. And now you have it too."
Meaning his sister actually used the force, as it says 'my sister had it' it shows a girlish shaped cloaked figure using the force.
In "Splinter of the Minds Eye" by Alan Dean Foster, she fights Vader with Luke's lightsaber, with ability that may be Force enhanced:
"But Leia performed a spinning, twisting arc in the air and brought her saber down in a slashing flare of blue light. Energy flashed as it contacted the Dark Lord's armored breath mask. Only superhuman reflexes enabled him to avoid the full effect of the blow."
Also here are three instances in three of the new Marvel Comics:
In issue #2 of Marvel’s Princess Leia, the ambitious young leader makes her way to Naboo on a mission for the Rebels, accompanied by R2-D2 and her pilot, Evaan Verlaine. Shortly after the trio lands, a mural of Queen Amidala stops Leia in her tracks. In three gorgeous panels, Padmé seems to come to life and turn to look at Leia with those kind, sad eyes. But when Evaan doesn’t see the same thing, Leia hesitantly dismisses it.
2. Shattered Empire - Leia feels the presence of Darth Maul through time.
Stronger with the Force than she was before, Leia experiences a much more intense vision than her first on Naboo. After the Empire surrounds Naboo with weather-altering satellites with the ability to cause horrific typhoons, Leia and Shara team up with flight-certified Queen Sosha Soruna to take on the threat. The Queen escorts Leia and Shara Bey into the same hangar that Leia’s parents once occupied during the Battle of Naboo, over 30 years before, and as she enters, Leia immediately feels the room go cold. She has a haunting vision of Darth Maul’s bloodshot eyes, his dark presence still lurking in the hangar decades after he killed Jedi Master Qui-Gon Jinn.
Star Wars (Ongoing) - Leia picks up and uses a Lightsaber.
The new run of Star Wars comics by Marvel have even given Leia the chance to pick up a lightsaber in one of the coolest massive fights since the Battle of Geonosis. While Luke spends his time complimenting Han for looking ultra cool with a lightsaber, our erstwhile Princess seems like a natural, especially with that strangely familiar arch of blue light emitting from the hilt.