Recovering a MyMonero wallet from seed
MyMonero uses a 13 word mnemonic seed. This is different from the 25 word seed used in the official GUI and CLI wallets. As so, you will have to go through a few extra steps to import your MyMonero wallet to the official local wallets.
As you may know, you only need your seed to access your coins. It can be used to generate the required keys to spend your funds that exist on the decentralized ledger (the blockchain). So as long as you have your seed or keys and they're not compromised, you're fine.
You can always restore your wallet with the mnemonic seed and access your funds. This also applies to the mnemonic seed used in the webwallet MyMonero in case it is down and you need to restore it with another client.
(You can also restore a wallet with a) the ".keys" file or b) private view and spend keys and the public address)
This is the 13 word "private login key" you were instructed to save when making your MyMonero wallet. Note: do not ever share your wallet seed or private spend key with anyone as they can be used to spend your funds. Keep your private keys private.
#1 Download Luigi's address tool
Luigi1111 is a core developer of Monero. He has made a nifty tool for addresses currently located in xmr.llcoins.net.
The tool looks like this:
If this looks intimidating, worry not, you'll only have to press a single button and copy-paste three things.
Download the tool directly from Luigi's Github repository here. The contents look like this:
Unzip it wherever you see fit and open the HTML document. If you try to directly open it from the .zip file it won't work as you need the javascript stuff in that "js" folder.
#2 Generate the keys
For both the seed and the keys: do not type them out, copy and paste!
Input your 13 word "private login key" to the Mnemonic Seed box and press the Generate button.
After pressing the button those 6 boxes will be filled with what looks like nonsense.
What we will need from those are the private spend key, the private view key and the public address.
To proceed with recovering your wallet, you'll need to have a CLI wallet setup. If you don't have a CLI wallet yet and don't know how to get it running check out the guide here.
#3 Make a new wallet
Open your command prompt and navigate to the location of your CLI wallet. Note that if you have downloaded the GUI wallet, the same folder includes the CLI wallet (monero-wallet-cli).
You should now have Luigi's tool with your keys and a command prompt window open.
To make a new wallet with the keys you just generated, do "monero-wallet-cli --generate-from-keys [walletnamehere]" without the "s. 
Note: if you want to use a remote node add "--daemon-address node.moneroworld.com:18081"
(monero-wallet-cli --generate-from-keys [walletnamehere] --remote-address node.moneroworld.com:18081)
As you can see, it'll first ask you if you want to restore from a specific block height. If, for example, you haven't received any XMR before 16/3/2016 you can use 1000000 here so it doesn't scan earlier blocks that don't contain transactions sent to you.
Next, copy and paste the keys as shown in the picture below. If you're on Windows, just right click and it'll paste the text from the clipboard.
That's it! You now have a new wallet with your moneroj from MyMonero that you can use with the official GUI and CLI clients.
If the wallet nags about a connection error, don't worry. Your wallet was still created and is in the same folder as the CLI client.
Simply exit (by typing "exit" and hitting enter) and restart with a remote node or start the daemon service and synchronize your own node!
Note that you won't see your XMR until the wallet has caught up with the blockchain. If you're connecting to a remote node this likely won't take longer than a few minutes. If you're synchronizing a full node from the ground up, it'll take some hours (if you have an SSD) or a few days (if you have an HDD).
It's always recommended to run your own node if possible as it is better for your privacy and for the health of the network.