This is to be filled in with recommendations for common parts (transistors, LED's, etc..) in an attempt to guide people and to make the possibility of bulk orders a bit higher. See also: Parts Selector.
MicroControllers
These are the ubiquitous microcontroller at the heart of the Arduino, Dorkboard, and many other hobbyist boards. This chip has a lot of resources, and is very inexpensive, making it a great value.
This is your standard, general purpose NPN (turns on when you apply a voltage to the base pin) transistor. It's in a TO-92 (3 pin, through-hole) package that's easy to breadboard. It can drive a maximum of 200mA. About $6 gets you a hundred of them.
This is your standard, general purpose PNP (turns on when you ground the base pin) transistor. It's in a TO-92 (3 pin, through-hole) package that's easy to breadboard. It can drive a maximum of 200mA. About $6 gets you a hundred of them.
A Half-Bridge (or H-Bridge) is good for driving motors with a very simple controls. The L293D is a dual-H-Bridge, capable of controlling two motors at the same time (or one four-pin stepper motor). The L293D has diodes that protect the internal transistors from the high voltages that motors can throw off.
This is a tiny ceramic capacitor used mainly for filtering out high frequency noise from a circuit. Common among many, many circuits, and $5 gets you 100.
This is an electrolytic (polarized) capacitor used to filter out larger gains and drops in current. If you're using a voltage regulator like a 7805, it's a good idea to have one of these next to it to filter out the ripple it produces.
This is a surface mount capacitor used for filtering out high frequency noise from a circuit. Common among many, many circuits, and $6 gets you 100 of them. "0805" is a good size because it's .08" long, which is most of the way between two pads in a through-hole protoboard.
This is a surface mount (0805 sized) ceramic capacitor. 100 for $6. "0805" is a good size because it's .08" long, which is most of the way between two pads in a through-hole protoboard.
This is an 8-bit "sourcing" shift register with latch. You send it data using three pins from your Arduino, and you can turn on or off its 8 pins. When a pin is "on" it's connected to power, putting out enough power to light up an LED. If you need more output than 8 pins, chain them together for tons more.
This is an 8-bit "sinking" shift register with latch. Like a normal shift register, you send it data using three pins from your Ardino, and you can turn on or off its 8 pins. When a pin is "on" it's connected to ground, allowing current to pass through it. If it's "off", it blocks current from going through it. This can also be chained together to control more pins.
These are tiny glass resistors whose resistance changes with the temperature. Measure the resistance (by hooking up another resistor as a voltage divider, and hook it up to an analog input for example), and you can measure changes in temperature very quickly. These won't give you an absolute temperature, but they're great for relative temperatures.
This is a digital thermometer which will give you a calibrated, absolute temperature as a 9-bit number, accurate to 0.5 degrees C for temperatures between -10C and +85C.
Press it, and it connects the pins on one side to the pins on the other side. For a more professional look, this button allows you to snap on a key-cap (see the digikey page for details.)
Under $0.10 a button.