Have a look at ModernDevice's 8x8 display, if 8x8 sounds like it would be fun. You can even drive multiple 8x8 panels.
http://moderndevice.com/8X8display.shtml
Look at the price! (But note that you also need a power supply... I'd take one from inside one of my old PCs!... and the LEDs. They aren't included in the kit because there's such a range of price/performance/color choices out there.)
And look at the .pdf document....
http://moderndevice.com/Docs/8x8LED_Instruct_0.3.pdf
(I have no interest in ModernDevice.... just a happy customer, and an experienced hobbyist who likes their apporach and support.)
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The answer to your question was in two parts...
a) The power demands of multiple LEDs... already answered
and
b) How can I drive more LEDs than I have pins for.
Assuming you meant more than driving 32 LEDs by connecting two to each of 16 pins, the answer is "multiplexing".
And there are two basix answers to multiplexing....
i) Software intensive
ii) Hardware "intensive".
In the latter, you add some electronics, chips, between your Arduino and the LEDs. Not hard to do, or expensive, and the wheel has already been invented... the 8x8 board mentioned above is one of several examples. It's the way I would go, for a bunch of reasons.
In the former, you would be doing what I'm about to explain, but probably on a larger scale....
You'd still need a little external electonics... transitors to drive the LEDs, to get past the power drain problem.
You'd fix up, say, four rows of five LEDs. In a crude system it would take 9 outputs from the Arduino to control the 20 LEDs. The "trick" is that you use four wires to control WHICH row of LEDs the other five wires are turning on or off at the moment. You then cycle rapidly between the rows...
First 10th second: switch the right LEDs in row 1 on or off,
Second 10th second, switch the right LEDs in row 2 on or off,
Next 10th second, switch the right LEDs in row 3 on or off,
Next 10th second, switch the right LEDs in row 4 on or off,
On the NEXT 10th second, you go back to row 1 again....switch the right LEDs on or off,
Next 10th second, switch the right LEDs in row 2 on or off,
etc,
etc
Because of "persistance of vision", your brain will not will "see" the LEDs flicker. Same thing that makes movies "work".