for moeyz, who asked an embarrassingly long time ago
Brief overview of household plumbing
Household plumbing breaks down into two systems: supply and drainage.
The cold water main, the first offshoot from the municipal grid (or well), branches into lines that separately feed the water heater/boiler and cold taps. Depending on local codes, the lines in your house may be of steel, copper, or plastic (PVC, ABS, or, in newer installations, PEX). Water pressure is determined mostly by the orientation of the greater water supply to your plumbing and can be modified by boosters and regulators. 50psi is good; 90psi damages fixtures.
The drainage system performs several functions. It is often abbreviated DWV, for drain, waste, and vent.
All household drains are followed by a trap, a U-shaped section of pipe that retains water. The retained water prevents sewer gases and small beings from entering your home through the pipes. Without a functioning vent, water is sucked out of the trap as the waste drains, defeating the purpose. Waste lines feature cleanouts, capped ports that allow easy access. Look for a square head plug.
Waste systems are gravity-fed and tend to be made with plastic pipe. Pipe that is not sloped to shunt waste to the sewer or septic tank must be attached to some kind of pump. DWV systems are not pressurized.
Tools and Their How & Why
Installing fixtures, pipe & fasteners
Ratchet & socket set, for installing and removing bolts. You want both standard and metric sizes, because appliance manufacturers often use metric bolts to make user-service inconvenient. A pretty good way to judge the quality of a ratchet is to make it click--smooth, even clicks indicate finer teeth that are less prone to seizing. Any brand name is acceptable. If you're buying used, be sure you're buying components with matching drives.
Pipe wrenches, for turning large pipe. You want 12", 18", and 24". Get aluminum if you can--it's quite a bit more expensive, but much lighter and easier to handle than cast iron. Flea markets are good places for aluminum pipe wrenches; try to spend about $15, $25, and $45, respectively. If you're buying used, be sure the teeth are good and sharp. That's a safety issue--those teeth bear a lot of leverage. To use a pipe wrench, fit it over the pipe with the jaws pointing in the direction you want to rotate. Orient yourself so that you are pushing the handle away from your face (ask me how I know). Pipe that is preferably left un-gouged can be protected with a rag or rubber sheet.
Channelocks, or slip-joint pliers, for turning small pipe and miscellaneous things. These work and are used in much the same way as pipe wrenches.
Adjustable wrenches, for the convenience of not repeatedly hunting down the correct socket size. Push these in the direction the handle is cocked, to place the load on the solid jaw rather than the adjustable one. The worm gears in adjustable wrenches are vulnerable to siezing; they are often held in place with a single screw and are easily removed and battered with WD-40.
Basin wrenches do not look like they do anything. They are used almost exclusively on faucets and the water lines leading to/from them. A basin wrench's pivoting head makes it applicable in areas very difficult to reach with other tools. One typically avoids unitaskers, but basin wrenches are sufficiently cheap and useful.
A Propane torch, to heat joints. Bypass anything with a speck of rust if you're buying used. If you're insane like me you might also batter any welds on the propane tank with soapy water and watch for bubbles.
Solder looks like thick, silver-colored wire on a spool. Use non-lead for pipes carrying water. Naturally, thicker solder is for larger joints.
Flux--derived from the Latin "fluxus", for flow--performs several functions. It cleans surfaces to be soldered, prevents the formation of oxides, and helps draw melted solder into the joint through capillary action. It is a thick grease sold in a small plastic tub. Apply a thin, even, complete coat before heating the joint. Note that different fluxes are manufactured for specific metals. Don't use acid flux on copper pipes.
Sandpaper, steel wool, and/or wire brushes. Oxidized metals solder poorly, being chemically different from non-oxidized metals. Removing all oxidation (inside and out) prior to soldering is the most important step.
Face/eye protection, to protect your face/eye. Solder drips.
A Coat, to protect your body.
A Blanket, to protect the floor.
First dry fit the pieces together to reveal whether anything is out-of-round. Oval-shaped pipe ends should be trimmed away. The fit should be firm, and without gaps. It will likely be necessary to support both sides of the joint as the solder cools; if you use wood, be sure to place it some distance away.
Once you're satisfied with the fit, pull the pieces apart, remove all oxidized material, and apply flux. Fit the pieces back together.
Keep the torch moving to ensure you heat both sides of the joint evenly. The joint itself must be hot enough to melt the flux; don't lay down hard flux and then melt it with the torch.
Unspool a few inches of solder and bend it into a curve a bit wider than the width of the pipe. Starting with the tip, lay it over the heated joint (you will suck at this the first few times you do it). Continue to heat both sides of the joint as before; solder moves towards heat. Note that it is possible to overheat copper; you will know you did so if the copper cools to a red color. Also, overheated copper is no good.
Let the joint cool for longer than you think you need to.
A tubing cutter, for small diameter metal and plastic pipe. The ratcheting kind that are used for PVC sprinkler line are not sufficient; they can't cut metal. You want the kind where the pipe is held against a jaw by a little round blade at the end of a screw. Use a tubing cutter by tightening said screw to about medium firmness and rotating the cutter in the direction of the jaws' opening, as with a pipe wrench. Rotate until you no longer meet resistance, tighten the screw, repeat.
A hacksaw and/or reciprocating saw, for sloppily cutting very small pipe and/or wood. Hacksaws are used in the same way as hand saws for wood. Cut on the forward stroke, pushing straight ahead (the cutting edge should be angled so this action applies the load). Apply no pressure as you bring the blade back, and do not pivot or twist the blade. Hold the work in place with your free hand.
Metal files, for de-burring metal edges. Notice whether the striations are parallel or cross-hatched. Files with parallel striations cut in only one direction--forward. Cross-hatched files can be drawn backwards as well, for faster work. Note that drawing the file in only one direction produces a cleaner edge; drawing back and forth causes an accumulation of tiny burrs on the face of the cut. It is also for this reason that you should draw away from the cut.
Rubber mallets, for hitting. A range of weights is useful to have on hand, but one-pound and two-pound heads should be sufficient. Beware the kind that are made of the same material as pencil erasers; those are prone to staining. Look for vinyl or some other non-porous material.
A level, if you wish to hang level pipe.
A plunger, for when you are still hopeful. Use quick, vigorous movements.
A hand auger, for clearing small drains. These consist of a wire wrapped in a metal sleeve with a ragged claw on one end and a crank handle on the other. They can be literally hand-powered or electric. Feed line until you meet resistance, turn the auger until (again) you meet resistance, then pull.
A closet auger, for clearing toilets and cleanouts. It is very similar to a hand auger: a stiffer cable with a hook on one end passes through a set of telescoping and rapidly-filthening sleeves.
A bolt/nut/washer set. Look for both standard and metric sizes. Because the unit-price on these is so low, and shitty fasteners are so very bad and inconvenient in both the short and long term, it's sensible to spend a little money on them.
An O-ring set, to repair small seals. They should not be stretched when installed. Tighten on them until they bulge out the sides just a little; if they start to scallop, you have tightened too much.
Teflon tape, to ease the installation of threaded fittings. Lay the tape down in the same direction you plan to twist the fittings. Make sure the foremost threads are taped, to prevent the tape from rolling over and gumming up during installation; replace janky tape. Five or six layers is good. Note that teflon tape does not seal anything.
Grease, for sealing threaded couplings. Apply to both male and female ends, over tape.
Caulk, for sealing. Apply an even bead that is generous, but not sloppy. Run a finger wrapped in a wet paper towel over the applied bead to seat it and remove excess.
Some Extra On Toilets
Toilets can run for a few reasons, being one of the more complicated devices in your plumbing system. If jiggling the handle to re-seat the flapper in the toilet tank doesn't work, you'll need to replace the fill valve, the flapper, or the flush valve.
The fill valve is the plastic cylinder-shaped assembly inside the toilet tank; it is attached to the water line with a nut, located on the underside of the toilet tank. Close the valve feeding the toilet, flush the toilet, and towel up excess water before disconnecting the water line. Be sure to put a new O-ring on the replacement valve. Adjust it so the cap sits above the water line, but not so it interferes with the tank's lid.
Rubber toilet flappers can harden and break down over time. They are inexpensive, and so easy to replace that explanation defeats the convenience of just looking at them and instantly figuring them out. As with replacing a fill valve, it's a good idea to close the valve to the toilet and clear the toilet tank, unless you don't mind putting your hands in that water.
In the second-worst-case, you'll need to replace the little wax ring that seals the flush valve, the assembly over which the flapper closes. With the tank dry, pull the old flapper off (it might be held with guide wires), clean the valve seat with a scotch pad or steel wool, and semi-firmly press a new wax ring over the valve seat.
In the worst case, you'll need to replace your toilet. This is slightly more complicated than building a small shelf from instructions. The tank is attached to the bowl with a coupling and a number of bolts, and to the wall via the water line; the bowl is attached to the floor with a pair of bolts and sealed with a large wax ring, which will be included with your new toilet. Mind which size toilet you get. It is a good idea to drain your old toilet before uninstalling it.
Clean the floor before centering the new wax ring over the opening. It's sensible to do a dry run, without tightening the bolts all the way or applying caulk, to ensure everything seats properly. You want to make sure the toilet flange sits half an inch above the floor, so the ring--being about half an inch thick--seals properly. Set the new bowl over the wax ring and tighten the two bolts, switching back and forth so you apply even pressure to the ring (this is a good rule of thumb when bolting together any gaskets). Apply caulk to the seam. In the exact reverse of what you did previously, attach the tank to the bowl and to the water line, via the fill valve.
At all stages, enjoy the extra money.