Your Dogecoin address is a string of letters and numbers which uniquely identifies your Dogecoin wallet (and hence, all of your Dogecoins).
What is it?Edit
The letters and numbers represent what is called a "public key", that the whole world knows about, and is used in conjunction with a "private key", which only you know. Doge addresses start with a capital D, followed by a number or capital letter. So, DFabcd1234[...] is okay, but Da[...] and cA[...] are not. Here's an article that describes everything in depth (it's about Bitcoins, but most of the concepts apply here too).
One key thing to keep in mind is: if you lose your Dogecoin address, you lose all of your coins. So keep up with it, maybe even make some backups (File > Backup Wallet, or "So Export") and save them on another drive, just in case.
Where do I find it?Edit
In your DogeCoin Wallet desktop app, there is a tab for Much Receive, which probably only has one item in it: "(no label) | D[...]". By right-clicking on it and then selecting "Copy Address", you copy just the address part to your clipboard, which you can use for other things (e.g. waterbowls).