Why doesn't my SSH key work for connecting to github?

Note: I'm not a newb, and I've done this a gazillion times, but for some reason today it decided not to work.

I keep getting the Permission denied (publickey). error message when trying to connect to github via SSH or when trying to clone a repo, even after remaking the ssh key and adding it to "SSH Keys" in my account.

This is what I tried to do ten times today without success:

make a key with ssh-keygen.
open ~/.ssh/id_rsa.pub with Gedit or Notepad++ and copy the contents.
Go to account settings on github.com
Go to SSH Keys
Click on the Add Key button.
give the key a title
paste the key into the key box.
Save the key (enter my github password to verify).

And now, when I try doing ssh github.com it just won't work.... What in the world? Am I just too tired right now or am I missing something?

Here's the output from ssh -vvv github.com

OpenSSH_5.9p1, OpenSSL 1.0.0f 4 Jan 2012
debug1: Reading configuration data /etc/ssh/ssh_config
debug2: ssh_connect: needpriv 0
debug1: Connecting to github.com [207.97.227.239] port 22.
debug1: Connection established.
debug3: Incorrect RSA1 identifier
debug3: Could not load "/home/trusktr/.ssh/id_rsa" as a RSA1 public key
debug1: identity file /home/trusktr/.ssh/id_rsa type 1
debug1: identity file /home/trusktr/.ssh/id_rsa-cert type -1
debug1: identity file /home/trusktr/.ssh/id_dsa type -1
debug1: identity file /home/trusktr/.ssh/id_dsa-cert type -1
debug1: identity file /home/trusktr/.ssh/id_ecdsa type -1
debug1: identity file /home/trusktr/.ssh/id_ecdsa-cert type -1
debug1: Remote protocol version 2.0, remote software version OpenSSH_5.1p1 Debian-5github2
debug1: match: OpenSSH_5.1p1 Debian-5github2 pat OpenSSH*
debug1: Enabling compatibility mode for protocol 2.0
debug1: Local version string SSH-2.0-OpenSSH_5.9
debug2: fd 3 setting O_NONBLOCK
debug3: load_hostkeys: loading entries for host "github.com" from file "/home/trusktr/.ssh/known_hosts"
debug3: load_hostkeys: found key type RSA in file /home/trusktr/.ssh/known_hosts:16
debug3: load_hostkeys: loaded 1 keys
debug3: order_hostkeyalgs: prefer hostkeyalgs: ssh-rsa-cert-v01@openssh.com,ssh-rsa-cert-v00@openssh.com,ssh-rsa
debug1: SSH2_MSG_KEXINIT sent
debug1: SSH2_MSG_KEXINIT received
debug2: kex_parse_kexinit: ecdh-sha2-nistp256,ecdh-sha2-nistp384,ecdh-sha2-nistp521,diffie-hellman-group-exchange-sha256,diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha1,diffie-hellman-group1-sha1
debug2: kex_parse_kexinit: ssh-rsa-cert-v01@openssh.com,ssh-rsa-cert-v00@openssh.com,ssh-rsa,ecdsa-sha2-nistp256-cert-v01@openssh.com,ecdsa-sha2-nistp384-cert-v01@openssh.com,ecdsa-sha2-nistp521-cert-v01@openssh.com,ssh-dss-cert-v01@openssh.com,ssh-dss-cert-v00@openssh.com,ecdsa-sha2-nistp256,ecdsa-sha2-nistp384,ecdsa-sha2-nistp521,ssh-dss
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-sha2-256,hmac-sha2-256-96,hmac-sha2-512,hmac-sha2-512-96,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-sha2-256,hmac-sha2-256-96,hmac-sha2-512,hmac-sha2-512-96,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: none,zlib@openssh.com,zlib
debug2: kex_parse_kexinit: none,zlib@openssh.com,zlib
debug2: kex_parse_kexinit: 
debug2: kex_parse_kexinit: 
debug2: kex_parse_kexinit: first_kex_follows 0 
debug2: kex_parse_kexinit: reserved 0 
debug2: kex_parse_kexinit: diffie-hellman-group-exchange-sha256,diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha1,diffie-hellman-group1-sha1
debug2: kex_parse_kexinit: ssh-rsa,ssh-dss
debug2: kex_parse_kexinit: aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,arcfour128,arcfour256,arcfour,aes192-cbc,aes256-cbc,rijndael-cbc@lysator.liu.se,aes128-ctr,aes192-ctr,aes256-ctr
debug2: kex_parse_kexinit: aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,arcfour128,arcfour256,arcfour,aes192-cbc,aes256-cbc,rijndael-cbc@lysator.liu.se,aes128-ctr,aes192-ctr,aes256-ctr
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: none,zlib@openssh.com
debug2: kex_parse_kexinit: none,zlib@openssh.com
debug2: kex_parse_kexinit: 
debug2: kex_parse_kexinit: 
debug2: kex_parse_kexinit: first_kex_follows 0 
debug2: kex_parse_kexinit: reserved 0 
debug2: mac_setup: found hmac-md5
debug1: kex: server->client aes128-ctr hmac-md5 none
debug2: mac_setup: found hmac-md5
debug1: kex: client->server aes128-ctr hmac-md5 none
debug1: SSH2_MSG_KEX_DH_GEX_REQUEST(1024<1024<8192) sent
debug1: expecting SSH2_MSG_KEX_DH_GEX_GROUP
debug2: dh_gen_key: priv key bits set: 122/256
debug2: bits set: 510/1024
d    ebug1: SSH2_MSG_KEX_DH_GEX_INIT sent
debug1: expecting SSH2_MSG_KEX_DH_GEX_REPLY
debug1: Server host key: RSA 16:27:ac:a5:76:28:2d:36:63:1b:56:4d:eb:df:a6:48
debug3: load_hostkeys: loading entries for host "github.com" from file "/home/trusktr/.ssh/known_hosts"
debug3: load_hostkeys: found key type RSA in file /home/trusktr/.ssh/known_hosts:16
debug3: load_hostkeys: loaded 1 keys
debug3: load_hostkeys: loading entries for host "207.97.227.239" from file "/home/trusktr/.ssh/known_hosts"
debug3: load_hostkeys: found key type RSA in file /home/trusktr/.ssh/known_hosts:16
debug3: load_hostkeys: loaded 1 keys
debug1: Host 'github.com' is known and matches the RSA host key.
debug1: Found key in /home/trusktr/.ssh/known_hosts:16
debug2: bits set: 497/1024
debug1: ssh_rsa_verify: signature correct
debug2: kex_derive_keys
debug2: set_newkeys: mode 1
debug1: SSH2_MSG_NEWKEYS sent
debug1: expecting SSH2_MSG_NEWKEYS
debug2: set_newkeys: mode 0
debug1: SSH2_MSG_NEWKEYS received
debug1: Roaming not allowed by server
debug1: SSH2_MSG_SERVICE_REQUEST sent
debug2: service_accept: ssh-userauth
debug1: SSH2_MSG_SERVICE_ACCEPT received
debug2: key: /home/trusktr/.ssh/id_rsa (0x14cce60)
debug2: key: trusktr@rocketship (0x14ce2b0)
debug2: key: /home/trusktr/.ssh/id_dsa ((nil))
debug2: key: /home/trusktr/.ssh/id_ecdsa ((nil))
debug1: Authentications that can continue: publickey
debug3: start over, passed a different list publickey
debug3: preferred publickey,keyboard-interactive,password
debug3: authmethod_lookup publickey
debug3: remaining preferred: keyboard-interactive,password
debug3: authmethod_is_enabled publickey
debug1: Next authentication method: publickey
debug1: Offering RSA public key: /home/trusktr/.ssh/id_rsa
debug3: send_pubkey_test
debug2: we sent a publickey packet, wait for reply
debug1: Authentications that can continue: publickey
debug1: Offering RSA public key: trusktr@rocketship
debug3: send_pubkey_test
debug2: we sent a publickey packet, wait for reply
debug1: Authentications that can continue: publickey
debug1: Trying private key: /home/trusktr/.ssh/id_dsa
debug3: no such identity: /home/trusktr/.ssh/id_dsa
debug1: Trying private key: /home/trusktr/.ssh/id_ecdsa
debug3: no such identity: /home/trusktr/.ssh/id_ecdsa
debug2: we did not send a packet, disable method
debug1: No more authentication methods to try.
Permission denied (publickey).

Solution

The GitHub ssh setup mentions testing your GitHub connection with:

$ ssh -T git@github.com

That follow the ssh uri syntax (also illustrated in "this answer").

But you did:

ssh github.com

(without any user). In that case, ssh reverts to the SCP syntax, which relies on a ~/.ssh/config file, with a section "github.com", to list:

the user
the hostname
(and optionally the public key location, but by default it will try ~/.ssh/id_rsa.pub)

To change it to a regular SSH URL, don't edit directly your .git/config file, as shown below.
Use the command git remote set-url:

git remote set-url origin git@github.com:username/repo.git

As noted by STEEL in the comments, should your private SSH key have a passphrase, you can add it to the ssh-agent with:

on Mac: ssh-add -K or (2023) ssh-add --apple-use-keychain
on Linux/Windows: ssh-add