It's internet money, like PayPal, but instead of PayPal keeping track of how much money you have, Dogecoin is kept safe with a list of secret keys stored on your computer or phone. Arf arf.
So it's like when you hand a dollar to someone, it has a serial number on it that the government knows is valid. Instead, when you send Dogecoin to someone, you prove that you have the coins by signing the transaction with your secret key, and then the Dogecoin network sees the transaction and agrees that it happened.
The difference is, there's no bank, and it's all anonymous. The only way you prove you have the coins is because your secret key can correctly sign the transaction.
Many devices like your phone or computer use something called encryption to make these sorts of proofs over the internet. Any time you visit a website starting with https, the server proves that they are who they say they are with a special key called a certificate that's owned only by that website. When the server sends data back, it performs a complex math equation on the data that only they could solve with their certificate, and they send an encrypted version of the data that only your computer can view.
The way these equations are designed, your computer can easily prove that the server solved it correctly and that they are who they say they are, but it's impossible for anyone else to ever solve the equation unless they somehow get access to the certificate. Much like your Dogecoin, it's only as secure as computers and servers are, which is why it's important that we have great engineers working on keeping our data safe.