This is NOT how to host a server with GoDaddy (to my knowledge, this is impossible), this is how to use your domain to forward to your home server.
First off you must have your home network set up to forward correctly to the computer hosting your server. There are plenty of other tutorials on how to do this.
Second, this only works if you have an "always on" connection as your IP address will reset every time you reconnect unless you lease a static IP address.
Third, this is only going to work if you have leased a domain name through GoDaddy (or you can figure out the commands for a similar domain host).
Step 1: Determine your ISP assigned IP address (for me I do this by logging into my router admin page and viewing the router status).
Step 2: Log into your GoDaddy account.
Step 3: Launch the DNS Manager (or whatever they're calling it today).
Step 4: Edit the zone for your chosen domain name.
Step 5: Add a new "A record".
Step 6: Change the host to 'minecraft' or whatever you want the subdomain to be. NOT minecraft.yourdomain.com, just minecraft.
Step 7: Change 'points to' to your IP address from step 1.
Step 8: Change TTL (time to live) to 1/2 hour (optional, but supposedly speeds things up).
Step 9: Click OK.
Step 10: Click Save.
Step 11: Twiddle your thumbs for a while (about a half hour).
Step 12: Try logging in with minecraft.yourdomain.com.
If your IP address gets changed you will have to update this with roughly the same process. You click the pencil by the record you created and change the Points To to your new IP address. Remember to click save!
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this can be done on most if not all hosting companies control panels, although the method may vary.
as mentioned in point 11 its not an instant update, it may take hours, but personally i have found godaddy to make the changes in under 20 minutes
I would like to add that you should delete anything else in the A Record too (just in the same section as you will put your IP, not anywhere else), unless you want your domain to alternate between parked and working (like mine did).
Maybe that's a given for some, but it wasn't for me so I assume it's not a given for everyone (unless I'm a complete idiot, which I won't discount).
Godaddy only allows for you to use dedicated ip addresses meaning 25565, so if you have a server ip with a port other than 25565, then it will not work.
1. You're completely wrong. GoDaddy doesn't handle the ports. it only handles the actual numeric address (128.0.1 for example, but not 128.0.1:25582). So, you can have a domain point to an address, but if you have a port that is not the same as the default used by the application (e.g. 25565 for Minecraft, 80 for a webserver) then you'll have to include the port. The port is ALWAYS handled by the application (in the case of an A record, but this changes if you are using an SRV record), NOT the DNS provider.