Why were they developed?
Bitcoin's creator, Satoshi, didn't answer this directly but in the paper on Bitcoin written by Satoshi, is the following:
While the [existing payments] system works well enough for most transactions, it still suffers from the inherent weaknesses of the trust based model. Completely non-reversible transactions are not really possible, since financial institutions cannot avoid mediating disputes.
So possibly, the entire reason for Bitcoin was to introduce a payments system of non-repudiable transactions (transactions cannot be revoked once issued and cannot be reversed.) There are many benefits Bitcoin brings, including an issuance level that is pre-determined (i.e., the currency won't be debased by unexpected "printing"). Other reasons Bitcoin might have been created:
I know they physically exist, from Ebay, but what's the difference with the Bitcoins on Ebay and digital ones?
A coin includes a private key and a public address. The physical coins such as what you see from Casascius (which is sometimes also seen sold on eBay) are simply a physical packaging for these two pieces of data -- the public address printed on the back of the coin, and the private key hidden under the tamper-evident hologram. Because of the cost of the physical packaging, manufacturing and order fulfillment, a physical bitcoin costs more to acquire versus the digital version stored in a wallet. If you wish to spend the physical bitcoin online, you peel the hologram and redeem the private key -- which only redeems the value of bitcoin(s) stored on the coin and the leftover physical packaging becomes worth (almost) nothing once the hologram has been tampered with.
Here's another bitcoin in "physical" form -- in the form of a paper wallet.
Can someone give me a BRIEF walk through on how I can take my cash dollars and get Bitcoins . . . and where those bitcoins can be stored.
You don't mention where you are. If you are in the U.S., the quickest method is to use BitInstant and visit one of their banking partners. You deposit cash, and get a redeemable code for that amount, less a fee (about 5%). That code then is used to buy bitcoins at the exchange you chose.
A cheaper method, in the U.S., is to use Dwolla (similar to PayPal), and link it to your bank and transfer funds to an exchange which accepts Dwolla (e.g., Intersango, CampBX or Mt. Gox). Then you pay only $0.25 to get your funds sitting at the exchange ready to buy bitcoins. This option takes around ten or so days though to get set up and get the first payment through.
Outside the U.S. your options vary, based on your location. Here's an article that might help:
. . . and where those bitcoins can be stored.
Some people store their bitcoins with the exchange. Some people withdraw their bitcoins so they are stored locally, using a client -- such as the Bitcoin.org client:
or StrongCoin). Some use a mobile client such as Bitcoin Spinner for Android. Some keep their funds on a paper wallet (like the one created by BitAddress.org above).