Nightcrawler is a mutant of many powers due to the fact that he is a second generation mutant. Here is a brief rundown of Kurt's powers:
[*] Ability to stick to walls
[*] Limited invisibility in shadows
[*] Enhanced night vision
[*] Enhanced agility and flexibility (there is some debate if this is due in part to his mutant physiology or strictly training, however)
Well, up above is really about all you need to know to follow the stories, but here's how each of those powers work, and thier limitations to the best of our knowledge.
Nightcrawler opens a portal or wormhole through a dimension adjacent to his own. Kurt passes through this dimension for a fraction of a second, and another opening is created up to 4 miles from his point of departure, Kurt travels between these two points near instantaneously. Upon his departure and arrival, a puff of smoke, flash of fire and the stench of brimstone accompanies him with a customary BAMF sound. This is atmosphere from the dimension he passes through escaping while the portal is open, and the BAMF sound is caused by the opening and closing of the portal.
On a couple of occasions, he has teleported considerably farther than his usual limitations. The most noteable examples of this are Uncanny #406/7 and Exiles #29/30. The Uncanny 'port was from just outside Paris France, to the Bavarian Alps, with the extreme fortune to land on an old friend's doorstep. In Exiles, he traveled from near Xavier's mansion to Central Park, NY. Both instances well exceeding his normal range. Both 'ports had two things in common. One, they were life or death situations that he was escaping from. In Uncanny #406, a bomb was rigged to go off in mere seconds, and he had to get away FAST, along with the unconsious Chamber. In Exiles, they were being attacked by Lobo's wolves, who had proven themselves extremely formidable foes before, and Kurt had to save Annie and the unconsious and badly wounded Carter, and get them to a hospital. Two, they were BLIND teleports. As in, Kurt teleported, but did not have a destination in his head when he did so. He just ported and hoped to end up somewhere. In both cases, he was extremely lucky with his destination. A bit TOO lucky for it to be mere chance, don't you think? Perhaps there is an explanation. One we can look to the issue #428 of Uncanny for precidence for. In that issue, Kurt was hurled over a waterfall as a mere baby, and before he reached the bottom, he was teleported away. Was this Kurt's doing? Not likely. Even ignoring that mutant powers do not normally develop until puberty, it is highly unlikely that an hours old infant would have the sense of mind to teleport himself to safety. No, it was his father, Azazel, who had saved him that day. Azazel, who has the same power of teleportation as Kurt, and happens to live in the very same dimension that Kurt teleports through. Azazel NEEDED Kurt (and all of Kurt's siblings) to live. Period. These teleporting mutants were the key to him escaping back to our dimension permanently, so he will do anything in his power to save any of thier lives. Kurt did a blind teleport. ended up in the dimension he teleports through because he had no destination in mind. Azazel opens an exit elsewhere, somewhere beneficial to Kurt.
There are a few side effects to his teleportation powers. Kurt has an infallible spatial awareness to help him gauge distances with uncanny accuracy, and guide him from one location to another. Kurt is capable of carrying passengers on his travels, but the ride is not plesant for them, and effects range from feelings of nausea to being knocked unconsious depending on the distance traveled. Kurt has learned to use this side effect to his advantage, grabbing foes in battle and 'porting around with them until they are unconsious. This is unforunately a drain on him as well, making him quite weak afterwards for a short while. But the effects on the opponent usually far outweigh any temporary weakness on his part afterwards.
Limitations: Kurt can teleport up to 3 miles north to south, and 2 miles east to west or straight up under normal conditions, but has been shown to go as far as 4 miles if he pushes himself to his limits.
The ammount of additional mass he can cary with him is limited, it started out that one person strained him to his limits, but through practice he has increased the additional mass he can take with him. It is now shown usually as up to 2 average sized adults, 3 if he pushes himself to his limits.
Kurt has to know precicely where he is going before he can teleport, either by being able to see it, or knowing the location he is going to very, very well. otherwise he could cripple or kill himself. The act of teleporting automatically displaces any gasses or liquids in his immediate area, but if he were to teleport into something solid, any part of him (or something he is holding) in the area of that object would become permanently materialized/merged in that object, which could cripple or kill him.
The velocity at which he was traveling upon departure form point A will remain constant upon his arrival at point B. So he can not reduce the speed at which he is falling, for instance. He can, however, change direction. So if he were falling at a great speed, he can teleport, change directions so that he is in effect 'falling' upwards, slowing himself to a safe speed with the help of gravity.
Ability to Stick to Walls
There are 2 popular theories going about to describe this ability, tiny hooks/suction cups, or connection to the dimension he passes through.
The first explanation to come out of Marvel was tiny little suction cups on his hands and feet which allowed him to stick to surfaces, much like gecko does. Personally, this never made much sense to me since he wore gloves and boots most of the time back then. but whatever.
The second explanation to come out of Marvel was that the dimensional wormhole, opening, whatever you want to call it, which surrounds him allows him to cling to surfaces. How exactly this works is a bit... well they never really said how it works, just that the connection to this dimension allowed him to do it, end of story. Some say he is able to affect bonding of surfaces on a molecular level. Others say he shifts gravity, manipulating the portal surrounding him so that the 'down' of that dimension corresponds to whatever surface he is sticking to, so that the walls or ceiling is pulling on him.
Limitations: in Uncanny 135, Nightcrawler could not stick to a frictionless surface because 'his fingers and toes couldn't get a grip'. Seems to me Claremont was writing with the suction cups theory in mind, but this was later changed. As of right now, I don't know if the frictionless surface limitation still applies.
Ah, the forgotten power. The invisibility power comes and goes based on Marvel editorial of the time. In the early days, Kurt was able to do it until it was decided he had too many powers, and one had to go, and the invisibility fell to the wayside. Later, it made a comeback after editors had changed, most notably in Excalibur. It appears the current editorial has decided this power is to make a comeback (again). he mentions it and uses it (tho not shown on panel) in Unlimited #49, and uses it in Uncanny #429, where it was clear he was not merely blending into deep shadows, but actually going invisible.
Just as with the wall-crawling, this has 2 explanations, one from early Marvel, one from later which connects the power to the dimension he teleports through.
The first explanation is that the colouriation of his fur is so dark that he tends to blend in with shadows, and that's that.
The second explanation is that the dimension he passes through when teleporting is always partially open, and it distorts, absorbs and/or bends light which makes it so that Kurt always appears in shadow even when in direct sunlight, and when in shadows it can allow him some degree of invisibility. This one seems to be the explanation adopted officially by Marvel, as more than once he has been shown to not merely blend into shadows but to actually become transparent.
Limitations: The degree of invisibility Kurt can acheive is directly related to how dark the shadows are. He can not go invisible at all in direct light, and can only go totally invisible in nearly complete darkness (which no one would be able to see anyway) so it is usually shown as about halfway invisible in medium-dark shadow.
Ability to See in the Dark
This was not stated as a power early on, but it makes sense and it came to be accepted as one of his powers later on. The explanation seems quite simple, his eyes are simply different than everyone elses. It's plain to see just looking at him, they are luminous yellow, sometimes even glowing. So, just like a cat or an owl's eyes, Kurt's eyes are simply better able to take in light and amplify it in low-light situations.
Limitations: While Kurt can see in very dark situations, he does need SOME light to be able to see and can not see in pitch blackness.
Enhanced Agility and Flexibility
Nightcrawler has been stated many times to have an extra flexible spine, allowing him to preform contortionist like feats and maintain a hunched over posture without damaging his back. So yes, his mutant phyiology DOES grant him additional flexibility. The jury is out, however, on enhanced agility. It is no secret Kurt is a superb gymnast and could do feats of acrobatics that would make your head spin. No one is doubting he is extremely agile. But is this agility due to training, or his mutant physiology? He grew up in a circus, he trained to be an acrobat from a young age, it wasn't like he woke up and could do these things overnight. But does his being a mutant help him with these feats? No one is really sure... I tend to think of it as intensive training for the skills, but aided by his enhanced flexibility, the additional balance granted by his tail, and perhaps slightly enhanced strength (which allows him to do much greater than average jumps)
The Dimension he is Connected to
Many people assume it is the Darkforce Dimension, which mutants such as the Vanisher, Darkstar, and Cloak use. But this fact has never been proven, and is mostly fan speculation. There was one fairly obscure bio of Nightcrawler published (one bio among several that call it an 'unknown dimension') that named the DF as the dimension he goes through. There was also one story a long time ago, Bizarre Adventures #27 which could MAYBE connect him to the Darkforce Dimension, but that could as easily be explained as the Vanisher's doing... I don't see a bio as proof, bios have been wrong before, and BA was very vague on the point of exactly which dimension they were in, though getting there involved Darkforce.
In the comics the dimension was until recently completley un-named, though it had been seen on a few rare occasions. Before Uncanny #430, It was shown most clearly in Excalibur #83, where it was shown as being filled with purple smoke and lightning, but that was only a brief glimpse and was likely only the upper atmosphere.
In Uncanny X-Men #430, we finally got a good look at this dimension, and it's inhabitants. It is a dry, desert like climate, with a turbulant atmosphere, inhabited by a race of mutants who once inhabited the earth, but were banished to this dimension as a punishment. Kurt's father, Azazel, is the leader of these mutants. Time moves slower in this dimension, which accounts for Azazel's extreme longevety.
Members of his own bloodline seem to have powers connected to this dimension as well. A link, it would appear, is passed down from parent to child, powers manifesting that utilize this dimension in one way or another. Nightcrawler inherited it from his father, Azazel, and he continues to pass it on to his offspring. Hence, Nocturne from Exiles can tap into it to cast her 'hex bolts', Blue in Days of Future Past inherited teleportation powers similar to Kurt's, as well.