Canon examples of Willow using telepathy in seasons 5 - 7
5.22 The Gift
Willow communicates telepathically with Spike across a battlefield. Note that this means Willow's telepathic powers apparently do work on vampires, unlike the ones Buffy acquired temporarily in 'Earshot'.
6.01 Bargaining part 1
Willow coordinates the activities of all the Scoobies using telepathy. Anya complains that it's 'intrusive', and Xander is very uncomfortable with Willow investigating "the lay of his brain".
6.21 Two To Go
Willow tells Dawn she can hear her thoughts if she "thinks real loud".
Willow communicates with Anya telepathically while in stasis, and with Buffy while raising the temple on Kingman's Bluff. After stealing the power from Giles, she apparently feels the thoughts and emotions of every living creature on Earth simultanously.
Willow hears it when Buffy "thinks real loud" at her, and relays a 3-way telepathic conversation with Xander.
Have I missed any?
Building on that information, what follows is my own personal fanon on what Willow's capable of and how it works. This may, of course, be superceded by new information revealed in the comics, but until then this is what 'my' Willow can do, and how I'd normally write her in post-Chosen fic. I've noted the canon support (if any) for my assumptions after each one.
How Willow's telepathic powers work
In seasons 5 and 6, she could use magic to acquire telepathic powers when she needed to. Once she absorbed the Coven's power from Giles in 'Grave', it became a permanent part of her; she was connected to everything and unable ever to turn it off again.
Based on the fact that Buffy, not Willow, initiated the telepathic conversation in 'Showtime'; and a general assumption from her speeches about being "connected to everything" both in 'Grave' and 'Lessons'.
Can sense other people's thoughts and feelings as a constant faint murmur in the background, like traffic noises or birdsong. If someone feels a sudden strong emotion, it's like a loud noise - and if someone thinks her name or visualises her in their mind, it draws her attention like someone saying her name out loud would.
This is based on what she does in 'Showtime': she was able to notice Buffy thinking at her. It's also what I assume happened in 8.03; she 'heard' Satsu's strong emotions.
One of the first lessons the Coven had to teach her was how to tune out the constant background noise through self-discipline and meditation, otherwise she'd have gone mad.
Gone mad as Buffy nearly did in 'Earshot', and how she herself did in the final act of 'Grave'. This is also my explanation for why Willow doesn't read everybody's mind all the time; if she tried to, she'd go mad.
Can speak into a person's mind, and read their surface thoughts if she concentrates. This can be difficult and confusing unless she knows the person well or they are visualising or verbalising their thoughts slowly and clearly. Cannot access people's memories or subconscious. She knows some spells or rituals that can do this, but it's not an automatic ability like her telepathy.
What I'm thinking of here is the spell she cast on Spike with Giles' Prokaryote Stone in 'Lies My Parents Told Me' - she couldn't access his subconscious memories using her normal telepathy.
The range of her telepathy is pretty much the same as normal human speech and hearing (or possibly, it works on anyone she can see). She may be able to use telepathy across longer distances as a special ritual. It does work on vampires and some demons, but not the more alien or powerful ones - and trying gives her a headache. Powerful magicians and other telepathic creatures may also be able to block her telepathy through their own powers.
This is based on 'The Gift' and 'Bargaining'. Saying that certain creatures can resist her telepathy is purely a storytelling device to stop her destroying every plot by instantly disovering the bad guys' motivations and secret plans...
Her ability to sense other people's thoughts means that she has first-hand experience of all the unpleasant and negative emotions - not to mention the inappropriate sexual attractions - that human beings can feel. At first this freaked her out. By now, she not only accepts it as normal, but it's helped her to reach a much more tolerant and philosophical view of human nature. At least, she knows the difference between normal anger, jealousy and resentment and outright Evil.
This is my own assumption, based on drawing logical conclusions and also on the Zen-like calmness she seems to have achieved by season 8.
Doesn't talk much about her powers because she knows that most other people would feel threatened by the idea of her prying into their thoughts. Remembers how upset she was when Buffy developed telepathy that time back in high school, and sympathises.
She showed much less patience with people who felt threatened by her telepathy back in 'Bargaining', but I'm assuming she's (re)learned a little humility since the end of season 6.
In fact, most of the new Slayers are simply overawed by Willow's power. They take it for granted that she should know everything about them, including their very thoughts. She's practically a goddess, after all; you don't resent a force of nature.
Another logical assumption.
Is pleased and grateful that Buffy does seem to be comfortable with the existence of her power, perhaps because she knows what it's like herself. Xander and Kennedy have more or less come to terms with the idea too, but Buffy's the only one who actually sometimes initiates telepathic conversations with her.
Buffy's casual acceptance of Willow's telepathy is based on 'Showtime'. That Xander and Kennedy would be less comfortable but still accepting of it is my assumption.
Occasionally has to stifle a sudden fit of giggles in the middle of a stuffy group meeting for no apparent reason. This causes everybody else to look at her puzzledly, apart from Buffy who hides an evil smile. Willow is secretly amazed at how imaginative some of the extremely rude mental images or cutting comments her friend keeps thinking in her direction can get.
Assuming Buffy takes it for granted that Willow can read her mind ('Showtime' again) I can well imagine Buffy taking mischevious advantage of this ability to tease her best friend.
Feels strongly that it would be unethical to reveal any of the things she learns from other people through her telepathy to anybody else, even Buffy. Would do so if it was a matter of life and death, but would still feel guilty about it.
Based on 8.03 'The Long Way Home', and more generally on Willow's strongly developed moral sense. (What? She has one. She just occasionally has problems remembering that it's supposed to apply to her as well.)
Since casting the Slayer Spell in 'Chosen', can automatically feel the presence of a Slayer within a hundred metres or so by sensing their aura. Concentrating and going into a trance extends this range to hundreds of miles, but with less accuracy. (She can tell if there's an undiscovered new Slayer in a particular city, but not pin it down to the district or street).
Based on 'Chosen' ("I can feel them, Buffy. All over. Slayers are awakening everywhere."), the abilities of the Coven, and my own extrapolations as to how this could work.
These Slayer auras are pretty much indistinguishable one from another, although thanks to long familiarity she can tell when it's Kennedy or Buffy, and usually when it's Faith. Dana's aura is also very distinctive.
Purely my assumption.
Thinks it may be possible to strengthen this bond to other Slayers into an even deeper connection, perhaps allowing her to see through their eyes, read their deepest thoughts, even transfer her consciousness into them, cast her magic through them, or meld with them the way she did with Xander, Giles and Buffy to defeat Adam. Isn't sure if this would work, and is reluctant to try it unless it were a life and death situation.
This is exactly what she does in 'The Long Way Home', but I'm assuming that she'd not actually tried it before, until it really was a life and death situation.
Can sense the nature and purpose of magic used in her vicinity instinctively, and get a better idea the longer and more carefully she concentrates on it. This can help her determine what the spell does and even how to counter it, break it or reverse it. However, she doesn't know everything, has been known to get things wrong, and can even over-confidently assume she fully understands a spell when in fact she's missed an important detail.
Based on her readings of Amy's magic in 'The Long Way Home'.