I have bought some bitcoins off local bitcoins and stored them in an Electra wallet in my computer. Given that I have now paid hard cash, how am I to determine that these are real?
Enter your Bitcoin address (the one you were paid into) into a public blockchain browser such as https://blockchain.info/. That will show you the relevant entries in the global shared Bitcoin ledger, which confirms that you, as the holder of the private key for that address, are the owner of those funds.
If you lose control of your private key for any reason, then your bitcoins might not be yours for much longer.
The best way to know your own private key is to go to a wallet generation site like bitaddress.org, and it will generate one of those key pairs for you. It will let you see both the public and the private keys.
If you use a wallet software it is difficult to see the private key; they are trying to make things less complicated for end users so they keep the key out of sight.
If you have your bitcoins in your phone, tablet or laptop, you can buy yourself some coffee in the nearest Bitcoin-accepting coffee shop. If they accept your bitcoins, you can feel confident they are real.
Alternatively, you can make a small online donation to help Ross Ulbricht. He's a Bitcoin martyr and a political prisoner. He is serving a double life sentence for creating the Silk Road, a free market that helped popularize Bitcoin in its early days.
Finally, you can also donate to a Bitcoin-accepting foundation:
Again, if any of these entities accept your donation, you can feel confident your bitcoins are real.
You can also transfer all your bitcoins to me. I'll check them for free and if they're real I'll transfer them back to you ;)