How to choose components for different applications:

Resistors - 

In most vintage amplifiers you will find brown resistors that are almost perfect cylinders.  Those resistors are carbon composite resistors.  Many purists believe that these resistors are a vital part to keeping a vintage amp sounding the way it does.  It may be true that the sound can be altered minutely by using other types of resistors, but I can't hear a difference. For most uses I stick to metal film resistors.  They are highly stable and the values are 1% values versus 5% for carbon film.  Mouser stocks XICON and KOA Speer metal film resistors.  I have used both with equal success.  The drawback to these resistors is that they are low wattage at only up to 1/2 watt.  For higher wattage applications I use metal oxide resistors.  Again I use the XICON brand and have been very satisfied.  Metal oxide resistors are 5% values and come up to 5 watts.  For larger resistor needs there are cement power resistors that go up to 25 watts and then Caddock power film resistors which go to 100 watts.

Potentiometers - 

A potentiometer is a device that allows you to vary a potential divider over a certain band.  They are sold in many popular values from 1K ohms to 1M ohms. There is then a slider inside that forms a voltage divider.  See below:

So just like resistors we have to take into account the wattage going through the resistive strip.  Most are rated around 1/4W and larger values get expensive quickly.  You can also use a pot as a variable resistor by only using an outside pin and the middle pin.  Depending on which outside pin you use, you will have a resistor that starts at the max ohms and goes down to 0 or the opposite.  

There are also two types, linear and log/audio.  Linear is exactly how it sounds.  As you turn a linear 1K pot at 1/2 the value is 500 ohms for both sides of the divider and at 3/4 one is at 250 ohms and the other is at 750 ohms.  Log or Audio pots operate on a logarithmic scale.  So as you turn the pot the value for R1 increases slowly at first and then rapidly near the end.  So at 1/3 turn the value is only 10 ohms and at 2/3 the value is 100 ohms, but it reaches 1K ohms in the last 1/3 of a turn.  Log pots are used on things like volume controls, as well as many of the other pots depending on design.

Capacitors -

So you have a vintage amp?  The capacitors in those amps are often the cause of many problems, including noisy hum.  The caps that smooth out that hum are filter capacitors.  These caps are notorious for going bad in vintage amps and will need to be replaced.  There are different styles for these capacitors, but all are called electrolytic capacitors.  They are special capacitors that are capable of high capacity and voltage in a small container.  The catch is that they can only do it in one direction.  Therefore these capacitors are polarized and will be ruined if installed backwards.  I use Vishay aluminum electrolytic capacitors, but many other companies make fine electrolytics.  There are also can capacitors.  These have multiple capacitors in one package and can make it more simple for design.

Capacitors are also used throughout the amp in the signal line to couple the stages of the amp together.  I use Sprague Orange Drop style capacitors.  These are very efficient and high voltage caps.  For small value caps I use silver mica capacitors by Cornell Dubilier.  It is vital to make sure your caps are rated at a voltage of the highest possible voltage that might go through them.  I would suggest the 600 volt caps since they will have a high enough rating and, since you are not buying them by the thousand, will not cost much more money.

Diodes - 

The diode is, of course, a check-valve for current.  So when we are looking at buying a diode we have to know the current that will travel through it and get one that will be able to handle the current.  It also must be able to withstand the given voltage so that they don't break down.  The mice thing about diodes is that they are dirt cheap...therefore I always go overboard on my diodes.  This can cause a small voltage drop, but the diode will never break down.  A 1N4007 from Vishay Semiconductors should be sufficient for most applications, but I will admit that I use a 1N5408 which is a complete over-kill, but even with these monsters I'm only spending $0.20 each!