I asked Exodus, probably the most powerful teleporter on the planet, to do a short piece on the hows and whys of this form of super-movement, but he said it was "boring", and Pyrepower wasn't willing to provide the "accommodation" he asked for in return for his services, so screw him. Instead, I'll talk about what is known (or suspected) about teleportation.
Larry Niven's "The Theory and Practice of Teleportation" gives us the obvious definition: "Teleportation is any method of moving from point to point in negligible time". Published in 1971, it included some speculation on psychic and mechanical teleportation. With the emergence of Apollo in 1962, it was hoped that provable psychic teleportation would eventually lead to a mechanical substitute. So far, no dice. Keep hoping, misfits.
Nobody is really sure exactly how teleportation really works. Most of the people who can teleport aren't interested in sticking around to be studied. It's one of the strongest arguments in favor of a radical new cosmology. Maybe we'll get there once a professional scientist develops the power.
If you can teleport, your lair is anywhere you want. Seriously, I don't even want to write this section.
That said, your best bets for lairs are abandoned salt mines, underground cave systems, abandoned military complexes, and that sort of thing. It usually takes some doing to find a secure location, so the hardest part about constructing such a lair will be research. Also remember that many such places will need ventilation or may have problems such as chemicals, radiation, or other contamination. To stay safe, educate yourself.
Slightly less exotic lairs can be constructed in hard-to-reach areas such as northern Alaska, Siberia, or the Antarctic.
Defenses against teleportation
We don't have force fields or weird alloys or anything that's likely to keep teleporters from entering a place. This leads to tightened sphincters from strategic defense thinkers and law enforcement types, and it's not likely to change in the future.
That said, there are ways to stop teleporters, but they're costly. Exodus was first caught because he blabbed about his next target to a cute girl who happened to be a Fed. They flooded the complex with an aerosol version of a "generic" (non-tailored) power-suppression drug, harmless to mundanes, and he finally took a Taser to the face once he actually teleported in and couldn't get out again. From what I hear, it cost them about $1.5 million, and they still couldn't keep him locked up for long.
Contrary to public opinion and popular fiction, teleporters don't necessarily need "line of sight" to hit their targets. Just how they reach their target varies from teleporter to teleporter, but an opaque wall isn't always going to work. That said, seeing where you're going is definitely safer.
Conservation of momentum and energy
Niven's article asserted that conservation laws must hold. Apparently they don't - teleporters can and have moved between altitudes or inertial frames of reference without difficulty.
I had a conversation with one teleporter (Papa Legba), who described the act of teleportation as "going out" and "coming in". He mentioned "the rushing of the wind that is not wind". When I asked him if you could "go out" without "coming in" again, he just laughed, and it was not a fun sound to hear. The impression that I got was that conservation issues are dealt with in the passage through the medium he hinted at - teleportation isn't a straightforward bridge between two points in space. Or that guy could just be off his rocker.
Teleportation as the ultimate killing tool?
Teleporters, generally, can't just pull someone's heart out of their chest. Current physics suggests that whatever teleporters do requires a clear energy differential - which is good if you want to teleport someone along with you and you do so by grabbing hold of them, but bad if you want to be selective. Apparently the "force" that teleportation exerts is stronger than gravity but weaker than intermolecular forces.
In layman's terms, a typical teleport can't pull your wallet out of your pocket, or your heart from your chest, but he can drag you somewhere if he grabs hold of you. Higher-powered teleporters, like Exodus, have been observed selectively teleporting objects away from people, even at extreme ranges - Exodus himself was able to pull a pair of pistols out of a man's hands from across the planet.
None of this really matters since you can just teleport a mundane 500 feet in the air and drop them, of course. Supers are another matter, but aren't we always?
The ability to rapidly teleport works a little like a speedster's powers, from what I am told - you basically "program" your mind and your subconscious does the rest. This is how Papa Legba is able to do his notorious "crossroads walk" and visit hundreds of places in a few minutes. Some teleporters have done similar tricks while in combat, moving rapidly and disrupting an enemy unit's formation.