The circuit for an LED lamp has only three parts: the LED, a resistor and a power source. These days, many LEDs actually already have a resistor built in, so you don't even need to add your own. The power source will vary depending on what kind of LEDs you are using but is generally a battery or batteries producing a small amount of power.

A circuit with more than one LED is made by placing the LEDs in "parallel" - if you imagine the battery wires as the sides of a ladder, the LEDs are the runs in that latter. Doing the circuit this way has a tremendous advantage, because if an LED fails (unlikely) or is broken, the other LEDs will continue to operate as before.

But I read that using just one resistor is bad for the LEDs!

I've read the same thing. The argument is that variations in the individual LEDs will cause variations in the current each consumes, making one LED brighter than the others. To be honest, that would certainly happen at higher voltage or current levels. But because I'm under-powering the LEDs (we only want a dim light!) that doesn't seem to be an issue.

How much power will I need? What kind of a resistor do I need?

Well, there are formulas to figure this out, but LEDs vary so much in their characteristics and are so tolerant of the low power designs we're using, there's no real point in trying to figure it out ahead of time; the "traditional" rule of thumb is that an LED circuit like this will need 3v of power and a 47 ohm resistor. But you may want to use a larger resistor if the LED(s) is/are too bright, or you may want to use a higher voltage if the LED(s) is/are too dim. In the end, you just need to feel comfortable with experimenting till you get the result you like.