Need help regarding power use for an Arduino
It is really simple to determine what sort of power source you will need. For starters, the Arudino boards have a built in voltage regulator, so they typically need 7V - 12V to operate. The microcontroller itself does not use much current unless it is driving things from the I/O pins.

These LEDs are controlled digitally by means of addressable driver chips. See this quote:


  The LEDs are 'chainable' by connecting the output of one stick into
  the input of another[...] There is a single data line with a very
  timing-specific protocol.


This means that you can control many of them in series (I don't know what the limitation is) from one single I/O pin. You could also split them into multiple chains controlled by multiple I/O pins. Your major limitations are CPU speed and the power supply.

The product page for your LED sticks says this:


  Each LED is addressable as the driver chip is inside the LED. Each one
  has ~18mA constant current drive so the color will be very consistent
  even if the voltage varies, and no external choke resistors are
  required making the design slim. Power the whole thing with 5VDC (4-7V
  works) and you're ready to rock.


Since each stick has 8 LEDs, it will draw at least 8 x 18mA = 144mA. Note, this power does not come from the Arudino I/O pins, but from a voltage regulator or straight from an unregulated battery.

Powering 10 of these sticks would require a power source of 4 - 7V capable of supplying about 1.5A (10 x 144mA = 1.44A). Powering 15 sticks would require about 2.2A. Of course, the power source needs to be able to supply this current plus a few mA more for the microcontroller itself (plus whatever is necessary for any additional circuitry).

Caution! Powering one or two of these sticks directly from one of the 5V source points on the Arudino board will work fine, but that voltage regulator can not power your entire array of 10-15 sticks safely. You didn't mention what board you are using, but on the Arduino Uno Board (schematic) the voltage regulator is an NCP1117 (datasheet). The datasheet says a current output of 1.5A (1500mA) is typical with a maximum of 2.2A, but I wouldn't suggest doing that for a prolonged period of time given the little consideration for heat dissipation on the Arduino boards.

There are a few ways to safely power everything:


Use two batteries: One 5V high current for the LED sticks, and one 9V for the Arduino board.
Use one 7V high current battery pack to power everything.
Use one 7+ V high current battery with multiple 5V regulators to power the sticks.


One thing is for sure: small batteries like a typical 9V alkaline cannot source that much current!



Edit



People seem to be confused about batteries, ratings, and capacities. I believe I have stated very clearly that common alkaline and the all-mighty 9V in particular CANNOT source enough current for more than one or two of these LED sticks for any prolonged duration. Actually, trying to draw too much from the battery(ies) at once will cause a severe drop in source voltage. Similarly, the voltage regulator on the Arduino board CANNOT supply a lot of current for long durations either, so using a powerful battery pack there will not suffice unless it also feeds the LED sticks directly, and the only voltage level I know of that could do that is 7.2V packs, but that might be too much for your LED sticks.

Your specs state the LED sticks have an input requirement of 4-7V DC. There are plenty of rechargeable packs on the market that fall into this span with high capacities and discharge rates. The capacity is how long the battery lasts, typically in mAh or Ah. For example, a 10Ah battery can (theoretically) supply 10A for 1 hour or 1A for 10 hours (and any other combination in between).

The discharge rate will specify the maximum (or sometimes most efficient) amount of current the battery can supply at once. So a 10Ah battery with a discharge rate of 5A can (at most) supply 5A for 2 hours. Get it? Good!

For 10 sticks, you need a discharge rate of 1.5A (although at least 2A would be much better for heat dissipation, overhead, and other such concerns). Normally, the sticks should be drawing about 1.5A in total, so multiply that amount (plus a little extra for headroom) by how long you want the LEDs to stay lit off of one charge to find your ideal capacity. Bigger is usually better, but know that batteries like to be fully discharged at least occasionally, and the bigger the battery  the longer it will take to fully recharge.