Controlling LED lights at a distance from controller

I am an artist that is making a model of an installation I have designed. I discovered LED lights and began researching how to use them. I have a vague hope that your Controller (R3) is the answer but am not sure. I would like to place multiple (20?), individually controlled, single LED lights in various positions on the floor of the model whose dimensions would be about 8x24 inches. I would also like to use some ribbon or strip LEDS that could be cut into pieces containing about 3 or 4 lights. I want each light to project a different color and be dimmable. I have looked at LEDs only on the internet as there is nothing I can find where I live to look at. So I am very naive. I can't quite figure out how everything is connected. I might need a single LED light or lights about 24 inches away from the control unit. I know I'm not making much sense but do you think the R3 unit will do what I'm asking. Maybe there are some tutorials you could recommend? I am also experimenting using key chain LED flashlights. I have one and it does what I'm looking for but only from the sides and not bottom lit. I think the full size installation will actually be easier than the model, however, I have to show what I'm proposing.

Thanks,

Burton

I can't quite figure out how everything is connected. I might need a single LED light or lights about 24 inches away from the control unit.

Leds are just diodes made from different materials (see second link below)

This link has a table correlating led color to led material combination types

The cathode must be negatively biased with respect to the anode and vice versa. (anode must be positively biased from cathode)
Also, the current must be limited with a resistor or it will self-destruct immediately upon power up.
The value of the resistance is a function of the turn on Forward voltage Vforward of the led , which is always listed on the package. Thus is the Vforward is 2.2V (dc) and the forward current (Iforward) is 20mA (0.020 Amps), then the current limiting resistor would be:

Let Vcc (Power Supply Voltage, (dc)) = 5.0 Vdc
Let Vforward = 2.2V dc
Let Iforward = 20mA ( 0.020 A)

RCurrent limiting = (VIN-Vforward)/ Iforward = (5V -2.2V)/0.020= 140 ohms

This is the maximum current for the led , not the normal operating current so in Arduinoland, the normal current limiting resistor value is 220 ohms , which yields a forward current of 12.7 mA minimum (although it could vary with the forward voltage)

In short , when you want to operate a led you need to know 3 things:
Vcc (power supply voltage in dc volts) (For example, to operate the same led as above off a 12V car battery:
RCurrent limiting = (VIN-Vforward)/ Iforward = (12V -2.2V)/0.020 A=490
** ohms**
VForward
IForward
RCurrent limiting = (VIN-Vforward)/ Iforward =

Since your an artist you probably will want to know the light intensity rating in lumens/per Watt
You may even want to the wavelength of the light in nanometers (nm) on the light spectrum scale
https://www.google.com/search?q=led+light+wavelength&tbm=isch&tbo=u&source=univ&sa=X&ei=INL4U5b6MMHcoATQ3IBI&sqi=2&ved=0CDQQsAQ&biw=1314&bih=652

Check out neo pixels since you need so few I would look at retail packs of them. There kinda high when compared to regular leds but you will find it much harder to control 20 dimmable leds from one microcontroler. There are ways to multiplex them but that's not for the beginner.

At the moment, if we disregard your inexperience and just talk about what would best suit your needs and let your figure out the details,
you should look into the
TLC5940
http://playground.arduino.cc/learning/TLC5940
, the WS2801
http://davidbu.ch/mann/blog/2011-09-29/arduino-and-ws2801-rgb-led-strip.html

, WS2803

http://forum.arduino.cc/index.php?topic=89104.0
, and WS2812 ICs.
https://www.sparkfun.com/products/11821
https://learn.sparkfun.com/tutorials/light-emitting-diodes-leds