Bouldering, which is short un-roped climbs done over crashpads with spotters ready to protect a fall, is the simplest and least gear intensive form of climbing. While bouldering can require extreme amounts of strength and control, it is relatively safe, because the only danger comes from landing incorrectly
Top-roping is the form of climbing most commonly thought about when someone hears the word "rock climbing" it involves the rope going up from the climber, through an anchor setup at the top of the climb, and back down to the belayer. it is the safest, and best kind of climbing for beginners, because the potential for long falls is almost completely eliminated.
Sport climbing is done by climbers who "lead" up rocks that are pre-drilled with bolts. Climbers clip "draws" into the bolts then clip the rope through the draw. More dangerous than toproping because when you fall, you fall twice the distance between you and the last bolt. Lead climbing is the only recognized form of "true" ascents.
Trad climbing, or "traditional" climbing is the origional kind of climbing. It starts with the climbers and their gear that the bottom of a cliff, and then they go up, the leader fixes placements, then the second follows and "cleans" the gear, or removes it from the rock. Very difficult climbing due to the fact that one has to place gear with one hand while hanging on to the rock with the other. Requires large amounts of gear and a lot of technical knowledge.
Unfortunately there are many wanna be climbers out there who talk like they are hardcore people but they wouldn't know an ATC from a Reverso, or Petzl from Black Diamond.
Real climbers climb because they love the rock, not for chicks or props or crap like that.
I was leading a 5.11B across a roof and took a whipper of the 5th bolt. I fell about 20 feet, and spiked my belayer up and his ATC got wedged in the first bolt.