I'm new to Bitcoin, how do I keep my bitcoins secure? With the recent rally and media attention, lots of new people joining the Bitcoin community.

A lot are asking: 

I'm new to Bitcoin, how do I keep my bitcoins secure?

and if they aren't that's the first thing you should teach them.
Use a desktop or mobile wallet (Electrum or Mycelium) to spend bitcoins (be sure to write down your recovery seed and store it like a paper wallet). To receive and store them, use offline paper wallets. As soon as a paper wallet exceeds some amount (50mXBC for instance), put it in storage and start a new one. If you are spending and your online wallet runs dry, break open a paper wallet to complete the transaction, any leftover coins will remain on the online wallet.
NOTE: with "online wallet" I mean wallets like Electrum and Mycelium, not web wallets like Coinbase or the Blockchain.info wallet.
Someone posted instructions for paper wallets on bitcointalk:

https://bitcointalk.org/index.php?topic=1559277.msg17075677#msg17075677

Do your own research before risking your coins following what someone told you.

Do not use web wallets.

EDIT: Brain Wallets (memorizing a passphrase that is hashed to produce keys and addresses) are highly controversial in the bitcoin community. The proper way to use them is to choose the passphrase uniformly and at random from a large set (at least 2^128).
Security improvements include:
Adding salt (something unique but trivial to remember, like your full name, phone number, or email address) to the passphrase to prevent hackers from attacking everyone at once.
Using key stretching WarpWallet is an algorithm that derives keys from a passphrase using a difficult, memory-intensive algorithm, this makes them much harder to hack.

To generate multiple wallets from one passphrase, you can attach an index to it (append a number to the passphrase).

The main advantage of brain wallets is that there is no physical wallet to steal or confiscate, and plausible deniability of having bitcoin.
Disadvantages include difficulty of remembering the passphrase (if you forget it, the coins are gone), possibility of theft (there are hackers constantly monitoring addresses created from weak passphrases, and they steal coins immediately), and the fact that in the unfortunate event of your death, your family would not be able to recover your coins.
If you decide to use them, they can be used in place of the paper wallets in the first paragraph.