Good question, and a common concern.  Yes, climbing is very safe.  Probably a lot safer than the car you get into every morning.  Indoor rock climbing is practically idiot-proof.  Initially, internal alarms blare a resounding...  "HOLY CRAP I'M GONNA DIE"  every time you take a little fall or are lowered after a climb.  But this is natural and an important safety mechanism.  In time you develop a working comfort zone within which you develop an understanding that if you follow some basic rules you are completely safe at all times.  These safety measures become as commonplace as buckling your seat-belt.  However when climbing there is always your climbing partner there to double check.  There is also a healthy margin for error built into the system, and consequently injuries in indoor climbing facilities are almost unheard of (there are occasional finger injuries and knee scrapes).  Big injuries don't occur as far as I know.  Outdoor is a different story, because it is based on highly variable set of circumstances.  When done properly, it is also extremely safe.

Top roping is the beginners method of rock climbing. That isn't to say that advanced climbers don't do it, but everybody starts with this method because its the simplest way to rock climb.  Top roping takes two people, a climber and a belayer, both with harnesses on. The rope connects to the harness of the climber, then to an anchor at the very top of the climbing wall, then back down to the belayer's harness.  The belayer controls the rope with a fancy little bit of engineering genius called a "belay device", I know... how'd they think of that one? The belayer is responsible for taking in the slack that is created as the climber moves up the face of the wall.    When the climber makes it all the way up to the top of the climb, the belayer gently lowers the climber back to the ground.  If it sounds complicated, its not.  I see 12 year olds figure it out and do it on their own in minutes.

There is a little bit more to it, like clipping carabiners and tying a knot... but it'll take mere minutes to learn.  Something else to remember is that the falls you take when top roping will never be more than a few vertical inches.  An attentive belayer will probably sense that you're struggling and cinch up those few last inches most of the time anyway.  I guess my point here is that top roping is not all that scary in the first place, but once you get used to it, its very comfortable.

Bouldering is about as simple and pure as climbing can get.  While beginners can most definitely learn to boulder, its slightly more difficult (or maybe just intimidating) than top roping.  Bouldering has no ropes, no harnesses, no nothing.  Well except for a pad that you can slide under you.  Yep, you just fall onto the pad.  But bouldering routes are short, under 12' or so.  So if you fall correctly its really not an issue.  Sure, there is the possibility that you could turn an ankle or something, but its really not that far.  Bouldering can be done alone or with a friend to spot you.  Spotters are often used to make sure you don't take a "bad" fall.  A spotter is someone that stands under but slightly out of the way when you're bouldering.  If you take some goofy swing and are about to land funny they make sure your head doesn't hit the pad first.  The goal in spotting is not to stop or slow a fall really, but mainly to guide a fall to the pad a little more smoothly.  If this sounds terrifying, reserve your opinion until you actually see someone doing it.  A 6' tall person falling from 10' only has 4' to fall, with a bend of the knees and a soft pad underneath you... its really very reasonable.

Lead climbing is similar to top roping, but not for beginners.  They make you take a test to see if you're capable before you're allowed to climb in that area of the gym.  The first big difference you'll notice, is that there is no rope hanging from a big top anchor like top roping.  There are a bunch of anchors scattered over the surface of the rock wall at about 6' spacing.  A rope is still used and you need a belayer, but you clip the rope into the anchors as you go up the face of the wall.  The scary part is that however high you get above your last clip you will fall double that.  Say you were 3' above your last clip, you'll fall 6'.  You still won't hit the ground, but it requires a different level of attention.  At this point you're probably asking yourself "why in the hell would you even want to lead climb?"   Well there are a few reasons.  When top roping, you generally have to stay under the anchor to avoid dangerous swings, lead climbing is not limited by that.  You can climb up 6' over 10' up 4' over 5' then up the last bit, this allows for significantly more elaborate routes.  Also, huge overhangs can be climbed without dangerous swings that would occur if you tried to top rope them.  And lastly, the very real possibility of a fall makes things a lot more REAL.  Its a bit of a mind game to be sure.

While outdoor routes are made by nature, indoor routes are created/designed by setters.  Setting is the process of selecting and placing holds in various orientations and locations in order to create a route that the climber is required to stay within as they progress up the wall.  Indoor and outdoor climbing routes both have ratings.  Ratings are numerical values that indicate the difficulty level (ex. 5.9).  While this is sortof an arbitrary judgment made by the setter, it is usually backed up by the opinions of others (sometimes its adjusted later).  Routes not only have different levels of difficulty but they vary greatly in how fun they are.  Some routes can be downright painful and annoying, some have unique balance moves, some put you into truly odd positions.  Some routes are relatively straight forward and some take a bit of finesse and perseverance to get through.  A good indoor facility has a good balance of difficulty levels, but also some really unique routes.  For the casual climber quality is important but frequently changing routes is not.  Frequent visitors are very aware of when they change the routes and itching to to try them out.

Lets face it, if you wanted to climb a ladder you could probably do that at home.  A good setter is that guy that you'll probably never meet, but makes the climbing experience both unique and worthwhile.  Challenging yet fun.  If all you've ever climbed is a "kids wall" at some fair, or a small one at some local weight gym...  I'm sorry to say, you've haven't really climbed yet.  You've put on a harness and experienced the sensation of height, but that's probably about it.