So it turns out my issue was that I was connecting my columns and rows together - d'oh! Its working great now, just waiting for a few more wires to arrive and I am planning on building this before Christmas: http://www.instructables.com/id/Yet-Another-Daft-Punk-Coffee-TableDisplay/

If you measure from C to 1, do you measure a short?

How would I go about testing this? If I do measure a short, what problem might this indicate?

Can someone help me understand how the matrix works? I've wired up a 3x3 matrix as in the attached image (no Max chip at this point). My understanding is that if I apply 3.3 volts to row C and make column 1 ground, the electricity should flow through the bottom-left LED (as in the top-right of the image), turning it on. What seems to be happening, though, is that the electricity bypasses the LED and flows directly to the ground, meaning no LEDs light up (as in the bottom part of the image). This makes sense to me since electricity usually follows the path of least resistance. But everything I am reading online seems to indicate that what I am trying should light up the LED. What am I missing here? Can anyone help me out?

I should also note that I have a 100 ohm resistor at the beginning of each row.

I guess the reason I just want to control individuals LEDs is that I don't have the wires, etc. yet to make my matrix, but I do have enough to plug in 8 LEDs and start experimenting with them. I'm hoping to be able to learn about the Max this way as I wait for the other parts to arrive.

Thanks for all the help so far. I ordered a bunch of stuff from Tayda Electronics - great prices. It has arrived, but I ordered some wires from another supplier and they aren't here yet so I can't build it yet. In the meantime I'd like to learn how to use the MAX7219. All the tutorials seem to be about matrixes, but I'd like to start by just controlling individual LEDs as I assume that would be more basic and give me a better understanding of how it works. Any suggestions on good resources for this?

"A" 74HC595 is not enough for a 5x5 grid, you need 10 control pins, and that only gives you 8.

Okay, thanks. I suspected that from a bit of Googling, but wasn't sure.

Spend $1.25 at taydaelectronics.com and get a MAX7219, a 30K trimpot, a piece of prototyping board,  and 2x20 pin headers and start building.
Will end up with something way more secure & reliable than trying to do it on breadboard. LEDs too if you need them.
Order today, will have parts in your hand in a couple of days.

Would this require soldering? I'm still trying to sort out how all these different components work. I guess I'm not really clear on how the pin headers work.

Thanks for all the help so far.

So basically build the matrix at the breadboard and then use a 2 pin connector to connect to each LED? I think that makes sense and could be a workable solution. I have a 74HC595 right now, does anyone know if one of them is enough to control a 5x5 grid? If not I might buy a MAX7219.

Not really the ideal solution because:
1. Price.
2. Flexibility. I'd like to have the LEDs a fair distance apart. I could use one breadboard for each LED, but having 25 breadboards for a 5x5 matrix seems a little unreasonable.
3. Breadboards are only connected in one direction (ie rows) whereas a matrix has rows and columns. You could make this work, but it becomes a complicated mess pretty fast.

I wonder if a metal clip of some sort might work? Maybe this? http://www.ebay.ca/itm/33mm-Silver-Prong-Metal-Hair-Alligator-Clip-20PC-Zh49-/320997036427?pt=LH_DefaultDomain_0&hash=item4abcea118b#ht_6343wt_1139
Or this? http://www.ebay.ca/itm/100-pcs-Silver-Plated-Extension-spring-connectors-Jewelry-findings-9-3-5-mm-/180991070573?pt=LH_DefaultDomain_0&hash=item2a23e8916d#ht_1560wt_1271

I'm wondering if there is any simple way to make a solderless LED matrix. I made a simple 2x2 matrix on my breadboard, but I can see that anything bigger will get to be pretty complicated in such a small space. Ideally what I would like is jumper cable that are in a y or some other sort of connector that would allow me to attach multiple jumper cables, but I can't seem to find such a thing anywhere. Does something like this exist?