A derailleur is a part on a bycicle that changes the gear by pushing the chain either up or down the sprockets.
It moves the chain sideways which shifts it from one sprocket/chainwheel to another.
The derailleur is the thing that the chain runs through, and which cause the chain to move from one sprocket to another when you change gears.
Derailer AKA derailleur is the mechanical thingy that moves the chain sideways to change gears on bicycles with external gear.
Cruisers are usually single-speeds, but can be fitted with either internal gear hubs or derailer gear hubs. Usually 3-5 speed IGHs or 6-7 speeds derailer gears.
On a MTB it's usually a clamp that goes around the seat post. On a road bike it can also be screwed on to a soldered tab on the frame.
Sure you can. But it won't gain you that much, as you will need either to keep the rear derailer or to install a special device to keep the chain tensioned.
First gear on a bicycle with external/derailer gears is with the chain on the smallest chainwheel(by the pedals) and the biggest sprocket(by the rear wheel).
No. Front derailers had to match the seat tube diameter and the pull direction, for rear derailers there's a 3-way match between the cable pull of the shifter, the throw of the derailer and the spacing of the rear sprockets.
You have to put your feet on the pedals... then push down to roll forward and move your feet in a circular pattern while steering with the handlebars. The clicking may be your chain rubbing on the derailer
Your derailer needs to be adjusted. If you look at the derailer from the side you should be able to see two screws that are used to set the limits of the derailer's movement. The one at the top determines how far towards the wheel the derailer will go and the lower determines how far towards the frame it will go. Lift the rear wheel of the ground, crank the pedals by hand and switch to the gear where the chain jumps off the sprocket, but stop cranking before it jumps off! Now tighten the top screw a little, and you should see the derailer move out a little. Get back to cranking with the rear wheel elevated and see if the chain stays in place. If it still jumps off, give the screw a few more turns. If it refuses to go onto the the sprocket, back the screw out a little. Do a search for "Park tools", "bicycle tutor" or "Sheldon Brown" for some nice online tutorials on bike maintenance.
On a bicycle, as seen from the right(drive) side, the chainwheel(s) and the sprocket(s) and the chain will be rotating clockwise. If the bike has a rear derailer the tension pulley will also be rotating clockwise while the guide pulley will be rotating counter-clockwise. If the bike has an internally geared hub the gears inside there can rotate in different directions (or not at all) depending on what gear the bike is in.
Well, first off the derailer usually haven't got any gears, it's just a lever that can move a certain distance sideways when something happens with the shifter. Secondly, that sounds like an adjustment issue. Either it's the limit screw for the RDs travel that needs backing out a couple of turns, or it's a cable length issue. Since I don't know if you mean the biggest or the smallest sprocket when you say "first gear" I can't provide any more specific help. I suggest you go to www.parktool.com, www.bicycletutor.com or www.sheldongbrown.com to read up on how to fix your bike.
Pulleys aren't that common on bicycles, pretty much only bikes with external gears have them. They're used in the rear derailer to let the chain change direction so that the derailer can take up the slack that comes from changing gears. Some ladies/mixte frames use a pulley up by the seatpost to change the pull direction of the brake wire. Yet some bikes use a pulley to get the right pull direction for the front derailer.
Sure. There are two things that has to match, the number of gears, and the actuation ratio or the rear derailer. If the rear derailer is a Shimano, then it'll take a Shimano shifter no problem. If it is an SRAM derailer, then you have to figure out if it's the Shimano-compatible version or the 1:1 pull SRAM version. If it's long-pull SRAM, you'll need a long-pull SRAM shifter. Once that's sorted out, unhook shifter cable at derailer. Pull grips off bar, loosen pinch screw(usually Allen head) slide twist shifter off. Slide trigger shifter on, tighten pinch bolt. Slide grips back on. Install shifter cable. Adjust shifting. Done!
Derailleur/derailer, cadence, peloton, drafting, velodrome,