I'm not an expert, but here's an explanation which hopefully is accurate enough:
Dogecoin is a digital currency system working as a decentralized p2p network.
You can store your dogecoins in a wallet on your hard drive.
There are also online wallets, for example
http://www.coinedup.com - you can log in there with a google account.
You can send dogecoins to someone else in transactions which are identified by encrypted code (you can send the code via the internet or even write it down on paper).
In order to prevent fraud, every transaction has to be confirmed (verified) by the network.
The confirmation process is done by solving cryptographical problems embedded in the transaction keyphrase.
You get awarded by coins for providing your computing power to the network (also known as "mining"). That's how new coins are distributed.
There is a fixed number of coins in the network. Currently, there are about 10 billion in circulation and this number will increase asymptotically until it reaches a maximum value of 100 billion long time from now.
If there are too many miners, the network automatically adjusts the "difficulty" of the problems needed to decrypt the transactions in order not to distribute more coins than expected.
Higher difficulty requires more processing power required to obtain coins.
The exchange rate of dogecoin is determined by supply and demand. Currently, the network is totally flooded with coins, that's why the price is so low.
Due to relatively low difficulty, you can mine thousands of coins per day (a few dollars worth) with an average PC, even more with a high-end one.
There are people out there who can obtain millions of coins per day with extremely powerful racks. Even if you subtract the electricity costs, it turns out to be quite a profitable activity
As more and more users join the network, it will get progressively harder to get new coins (a good thing) and the price will increase. But if the interest is lost, the coins will remain easily obtainable (a bad thing) for those who stick around, negatively affecting the value.
Hope that clears things up a little bit