In Star Wars: The Force Awakens, when
Han Solo died,
Leia became aware of it instantly.
Yes, Leia is force sensitive, but when Alderaan was destroyed, Obi-Wan heard billions of voices crying and going out of existence which is generic. When
Han Solo died,
Lots of people were also dying there. How could Leia intercept particular death signal provided that
Han wasn't force sensitive.
Talking about love, when Padme died, Vader never sensed it. Then, how come Leia sense this?
In @SS-3's comment is perhaps another way to answer the question.
Edit: 1) Additional supporting material (re: Kylo Ren) from the novel was added to provide a more comprehensive illustration of the two-way 'action and reaction' between Kylo Ren and Leia. 2) Caps usage has been trimmed down.
1) Leia knows that Han is going to attempt to
'reach' their son.
This is a fact as this is something she specifically requests of Han before he leaves as she tells him to
"bring him home".
2) Those sensitive to the Force have the potential to sense others who are also sensitive to the Force... (or sense the deaths of many like with the destruction of Alderaan, for example).
This is a fact re-suggested in the movie when the Supreme Leader and Kylo Ren talk about sensing Rey who is sensitive to the Force and is awakening to it.
With these two ideas in mind, my answer is that
Force-sensitive Leia sensed the abomination that Kylo Ren - her own force-sensitive son - did, which was to kill his own father.
I believe this answer is potentially further supported by two particular instances in the book where:
1a) Kylo Ren reacts to the mention of his mother...
“No, it’s not.” Halfway across the walkway now, Han continued to move forward, smiling. “Never too late for the truth. Leave here with me. Come home.” Without the slightest trace of malice or deception, he cast a dagger. “Your mother misses you.”
A strange sensation touched the younger man’s cheeks. Something long forgotten. Dampness. Tears.
“I’m being torn apart. I want—I want to be free of this pain.”
1b) Kylo Ren reacts to what he's done...
Stunned by his own action, Kylo Ren fell to his knees. Following through on the act ought to have made him stronger, a part of him believed. Instead, he found himself weakened.
and
2) Leia admits that she knew of
their son's potential corruption by and to the Dark side since the beginning... and yet tried to deny it and hide it because 1) she didn't think Han would handle it too well, and 2) she wanted to believe that their son could be kept with the Light and either not turned or capable of redemption.
In this, Leia shows
denial - YEARS of denial - when in truth, she knew deep down and long ago what really was going on and was likely to happen.
Sending Han out to try and save their son reads, to me, like another last-ditch effort where she - once again - is 'naively' hoping for something that will likely not happen the way she wishes to.
Though we have no idea as of yet if
redemption is in the cards for Ben, killing his own father
is a pretty big deal and coupled with the fact that
Kylo Ren is shown to emotionally react to the mention of his mother (or just the sense of family)...
For Leia, I think that's more than enough of a 'disturbance' in the Force
(specifically, her son emotionally reacting towards the mention of her and family and being corrupted/turned enough to the Dark to kill his own father)
for her to sense and be aware of.
Note: Please excuse the choppy formatting of this answer as I am trying my best to ensure the majority of spoiler-worthy material is kept behind the spoiler tags while still keeping my answer readable.
Firstly, the timing isn't quite how you've portrayed it in your question:
Leia reacts to Han's death well before the planet is actually destroyed, more or less immediately after he is killed.
There is also precedent for Leia being sensitive in this way,
i.e. able to sense the well-being of those she is very close to.
When the second Death Star is destroyed, Leia is confident that Luke escaped the explosion.
Han doesn't need to be a force-user for this kind of sensitivity to work. In The Empire Strikes Back, Vader's troops torture Han so that Luke will sense that his friends are in danger and attempt to rescue them, being drawn into Vader's trap.
Another good example of this kind of sensitivity is Yoda being able to sense the jedi being slaughtered during the execution of Order 66. Many of the jedi are dying in war zones with lots of other death around them, but it's strongly implied that Yoda is sensing their deaths specifically rather than just general carnage.
Regarding Vader/Padme, Padme's death occurs after Darth Vader loses his fight with Obi Wan on Mustafar, and before he wakes up from the life-saving medical treatment that leaves him in his famous suit. He was either unconscious or wracked with extreme pain at the moment she died, and so far this force-mediated awareness seems to occur at the moment of death rather than hanging around afterwards. (It's also possible that he wasn't well enough recovered to trust his senses, or that he sensed she was dead but was in denial about it.)
The novelization by Alan Dean Foster explicitly indicates Leia felt a Force shudder when
Han
died, NOT when other people died.
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.
As to "how", @Renegade Princess answer offers some compelling theories, and i'll add one more of mine:
He was killed by her son, a strong Force-user Ben/Kylo Ren. If nothing else, she likely could have sensed the event due to that connection, even if you're unwilling to accept that a Jedi can sense the death of non-Force-sensitive individual, despite that answer's arguments.
The same way Luke knew in 5 that his friends were in trouble, The Force.
YODA
(shaking his head) Hmm. Control, control. You must learn control.
LUKE
I saw... I saw a city in the clouds.
YODA
Mmm. Friends you have there.
LUKE
They were in pain.
YODA
It is the future you see.
There are also other times when they know things through The Force, like when Vader tells Luke to search his feelings to confirm that Vader is his father, when Obi-Wan senses the destruction of Alderaan, Leia confirming that she and Luke are siblings, etc.
In the Legends stories, it was clear that people use The Force in different ways, so we could have said that Leia had the ability naturally or had developed it.
Now that we only have the movies, we can see that it is an established principle (The Force might tell you if something bad is happening to someone you care about), but we do not have enough details on how The Force works to say for sure if The Force will always tell you, how much you need to care, if you can hate the person for the effect, how easy it is to ignore, how detailed the information is, etc.
As for why Leia and Luke did not comment on the destruction of Alderaan like Obi-Wan, Leia was there and knew what was happening, so there was no point in commenting on it. Since she was not Force trained and in the middle of a traumatic experience, she probably did not recognize The Force. Luke was also untrained and was focused on his lessons.
