Ropes / webbing: 
Early in mountaineering, the natural fiber ropes were so likely to fail, the rule of the day was "The leader must not fall!" The development of synthetic materials, principally nylon, has now produced ropes that can safely hold almost any fall, are engineered to absorb shock and reduce the impact on the climber and the anchors. They are lighter, impervious to moisture and rot, and much more durable. Even so, ropes to require both care and careful use. They need to be washed when dirty, stored properly, and kept away from sunlight and chemicals. In use, they need to be protected from sharp edges, rockfall, and abrasion. Typical duration of a ropes life depends on use and may very from one hard season to 5 years. Ropes are generally retired from climbing use after 5 years. 
The development of flat and tubular nylon webbing has been of great use for climbers. Very strong and resistant to abrasion, it is typically used to build anchors which direct the rope away from hazards. The nylon webbing is sacrificed to take the abuse instead of the rope as it is much more inexpensive to replace. 
Shoes: 
The development of sticky rubber compounds in the 1960's led to an evolution of climbing shoe designs from mountaineering boots into the modern lightweight models seen today. The new rubber allowed climbers to challenge previously unclimbable routes and push the limits of modern rock climbing. Climbing shoes are one of the first items acquired when you start climbing. Shoes are a very personal item requiring a precise fit to get the most out of them. They are designed to focus the power of your feet where needed. There should be no sloppy play or looseness, fitting from snug to tight. There are many designs available, you'll want to find one with good fit and the performance you desire. Climbers often own several pairs of shoes, specializing for different types of climbing. 
Climbing shoes are not made for walking, and are usually quite uncomfortable to wear when one is not climbing. Additionally, the soft and sticky rubber will wear out and pick up dirt much quicker. They are usually put on just before the climb, and removed soon after. Approach and retreat is done in a second pair of shoes appropriate to the terrain. Fortunately, when the soft rubber soles wear with use, they can usually be replaced with fresh rubber soles as long as the uppers remain intact. 
Harnesses: 
We've come a long way since the early practice of rapping a few lengths of rope around the waist and tying it off. Not only did this eat up precious rope length, but it put quite a shock on the climber if he fell. Imagine the force on your lower back that would come with even a moderate fall, ouch! When nylon webbing came on the scene, climbers built harnesses that spread the load over wider areas and incorporated the legs into the system of the modern "sit harness". Padding was added, and loops to attach gear. A good harness can last a lifetime, choose one you like that suits what your climbing style. The best way to judge a harness is to hang in it a while and see how it feels. Compare several until you find the one that fits you best. 
Protection: 
Protection refers to methods or hardware to anchor a climber to the terrain. At first, the only protection for a climber was that which came naturally - a sturdy tree, a loop of rope around a boulder, or slung over a rock horn. Savvy climbers would wedge chockstones in cracks and tie off to them. When many people think of rock anchors, they envision mountaineers of the 50's and 60's pounding iron pitons into cracks with large hammers. Many modern day classics were established this way. 
Though pitons and hammers still have their place in certain instances, for the majority of climbing they are no longer used. This is primarily because of "environmental" concerns. Hammering home a piton destroys some of the rock, as does removing it later. What may have started as a tiny knifeblade fissure would gradually expand to a larger and larger crack, eventually perhaps becoming large enough to be used as a hand or foot hold. This changed the very nature of the climb, altering the experience for subsequent parties. Climbers looked for better ways to climb. Experiments with non-destructive methods included tapered wooden "chockstones", and threading or wedging knotted webbing through cracks. Next came large threaded hex nuts reputedly found in a rail yard and slung with webbing. This trick proved surprisingly effective, and lead to the development of much of the basic "static" protection we use today. 
Todays climbing "pro" can be divided into two categories, static and dynamic. Static just sits there, dynamic uses mechanical advantage to transfer force into holding tighter via moving parts. Static protection is commonly called nuts, wedges, chocks, or hexes, hinting at it's train-yard hardware heritage. These metal wedges are slung with wire or cord, and fitted into wedge shaped cracks in the rock (when they can be found.) In the 70's, western climbers found themselves confronted by beautiful long parallel-sided cracks, wonderful for climbing but impossible to protect. Dynamic protection emerged, ideal for those parallel cracks. Dynamic devices are commonly called "cams" or camming devices, as the typically use an eccentric cam on a single or double axle which is forced to expand and tighten in the crack as pressure is applied. (Just remember "cams"). 
Just which pieces of protection to bring on each climb is part of the game. Typically a "rack" of gear will include a general range of sizes with a mix of nuts and camming units. Specific pieces are then added or removed in accordance to the climb. You wan't to bring just enough to do the job, without lugging excess gear along with you. When and where to place which piece of gear, quickly knowing which one will work, selecting and placing it solidly, and familiarity with how to retrieve it are skills that take time to develop.