Probably the best known user of phasing powers is the hero Covenant, and I can't interview that guy. So here's some speculation and second-hand information.
Important note for active supervillains: contrary to reports, I believe that the Icons member Diver does not have phasing powers, despite her attempts to cultivate the impression that she does. Apparently it's all an illusion generated by her light and invisibility powers, so don't be fooled!
Phasing is the act of moving through solid matter without disturbing it - basically, you're a ghost. I was looking for a better name for this, because supers with phasing would have to be called phasers, and this isn't Star Trek. Oh well, screw it.
This is one of the more subtle powers, because not everyone has an opportunity to see you do it. If you are able to phase (I'm not saying phaser here, people!), and you can get away with not revealing it, try not to - it makes a fantastic trump card.
Defenses against phasing
The physical basis for this power is poorly understood. The Transhuman Capability Catalog tentatively calls it a subclass of electromagnetic control, based on the theory that the super is flooding the atoms of his body, and whatever he's passing through, with enough energy to move their electrons into higher orbitals and give the Pauli principle room to work. There's some quantum weirdness past this point which I don't feel qualified to discuss.
This opens the door to some sort of anti-phasing defense, though. Some sort of metamaterial where electrons are already super-charged, or a macro-molecule (a large carbon nanotube for example) where the atoms' electrons are already in a fixed energy state, might resist phasing. I'm not aware of anybody that has actually deployed this sort of defense, but if you hear of one, let me know.
Moving and other considerations while phasing
In general, people don't worry about falling through the floor while phasing. Their bodies develop a mechanism to push off of nearby atoms electromagnetically while in the phased state, and their subconscious mind takes care of doing what their conscious mind seems to want. They can walk up or down stairs, "climb" through solid material, then step out onto a solid support.
From the interviews I've read, people have described it as feeling like swimming. Virtually every phasing-capable super (still not saying it) has developed an adaption for breathing, and some have shown adaptions to high-pressure conditions, the way aquatic supers have.
As a defense mechanism, phasing isn't perfect. It takes effort and energy to maintain the phased state, so weaker users (not saying it...) must still defend themselves from bullets and such in the usual way - by not getting shot. Stronger supers can just stand there and let the lead go through their bodies.
I have yet to see a battle between two such supers, so I have no idea how their powers will interact. That said, I'm sure it'd be interesting.