Is Lightning running on the Bitcoin protocol or on another blockchain? Is is something like a sidechain? Where are the bitcoins in the Lightning Network stored?
Is lightning running on the bitcoin protocol or another blockchain?
The Lightning Network operates on the Bitcoin blockchain. However it has its own network protocols and protocols with transactions. It merely uses Bitcoin transactions.
The Lightning Network can be ported to different blockchains so long as they support a similar feature set to the Bitcoin blockchain. For example, the Lightning Network has been successfully used on Litecoin too.
Is is something like Sidechain?
The Lightning Network is not a sidechain.
Where bitcoins in network are stored?
They are in transactions, just like everywhere else. There is no such an object as a Bitcoin. Rather a "Bitcoin" is just a value in an unspent transaction output. Those exist on the blockchain. When using the Lightning Network, the Unspent Transaction Outputs for your Bitcoin are still on the blockchain. The main concept of the Lightning Network is that the transactions that spend those UTXOs are not broadcast to the Bitcoin network immediately but rather kept private between two parties who are transacting with each other. It is only broadcast when the two parties no longer wish to transact with each other.