The coins are "stored" in the blockchain.
Electrum holds the keys that can "unlock" your ability to spend bitcoin you have received.
Electrum (and every other wallet) DON'T store coins. They only store keys. That's a common misconception.

In order to "receive" coins, you need NOTHING at all. Just give someone an address.

You need Internet in order to:
1. Monitor your incoming transactions and perhaps get notifications about them.
2. Broadcast transactions a.k.a "send coins"

You don't need Internet to:
1. Receive coins
2. Sign transactions (assuming you will broadcast them layer)
3. Create new wallets aka new pairs of keys