Using Electrum and other Bitcoin wallets is pretty safe (assuming your computer isn't hacked). But if you're serious about security or want to keep some funds portable, you can use a "paper wallet." The security benefit here is that your private key isn't connected to the internet, so no hacker can get to it. When a private key is stored offline (either on paper or in a non-connected computer), it's considered "cold storage."
A paper wallet is just a public and private key printed out. That's it. All those companies that offer you pretty green fancy-looking paper wallets with holograms are silly – all you need is the public and private key. (It's best practice to encrypt your private key as an added layer of security. We'll get to this below, so be patient. Jeez, why are you so pushy?)
The cool thing is that you don't need to be online to generate a public and private key. In fact, doing it offline is the safest way to generate your keys and be sure nobody's snooping on you. This brings us to an amazing little tool called bitaddress.org. You can actually download the entire site as a single .html file, which you can run in a web browser while disconnected from the web. To download it, visit bitaddress.org, click the GitHub link on the bottom, and click the "Download ZIP" button on the bottom right.
As a sidenote, you should be skeptical about trusting the author of this tool (or any tool that generates keys for you), since the author could theoretically generate tainted public/private key pairs they could steal from you in the future. I researched this tool a lot, and apparently enough security people have examined the code and nothing looks nefarious. Also, I found the original thread where the developers are talking about it here . Pretty cool.
Ok, so once you've downloaded the bitaddress html file, open it in a browser. You'll see you have to move your mouse around to generate true random numbers. This is pretty badass.
After you've moved your cursor around like an idiot for a few moments, the page will load. Click on "Paper Wallet."
By default, you see those stupid-looking faux-currency bills. Fuck that. They take up too much space on your printout and are a distraction. Select "Hide Art" and generate 7 Addresses using BIP38 encryption, like this:
Then, in addition to printing this page, I like to make a PDF and keep it handy. Since I'm still learning/practicing with Bitcoin, it's much easier to copy/paste the addresses from a PDF than it is to scan the QR code and email it to myself.
So now you're free to send BTC to any of the newly-created Bitcoin addresses in the left-hand column. And you can monitor the Address's balance by searching for it on blockchain.info.
To learn how to retrieve the funds in a paper wallet, check out my next post appropriately called "Importing Bitcoin from a paper wallet into Electrum."