see bro is like this:

private ip and public ip used by maxis mean this...

when u use PRIVATE IP

UR HOME NETWORK (ur router) >---------> THE WORLD
means... the world see IP to ur network directly
when u use PUBLIC IP.....

UR HOME NETWORK >------------------> MAXIS PROXY >------------> the WORLD 
means... everything u request.. go to MAXIS first... then MAXIS ask for it
the world never see IP that directly connect to U... they see MAXIS IP

Maxis say this for better security coz hackers cannot directly attack... but disadvantage is u share IP with many ppls (coz its MAXIS IP)... so if u downloda from free download site, they tell u ur IP ALREADY DOWNLOADING 
OR u wan host game or website at ur home... cannot also.. coz no IP directly to ur network

Lol I hope u aren't confused by static dynamic public and private.

Static is fixed, that means u cut power unlimited times and reconnect unlimited time, u get the same ip, dynamic just all random. So if you own 123.123.123.123, u always get that number, u need to pay for static ip and they are expensive and will be more and more expensive. Normal Internet user get dynamic ip, each time they connect, they get different number. What is the different? If u have fix, u can host things, ppl know you are 123.123.123.123, ppl can directly find you. If dynamic, u tell your ip to somebody, the next time your router reboot or reconnect u get different, your old ip goes to somebody else. Check www.whatismyip.com

Private is internal ip, public is external, u cannot choose to have no public ip as others unable to send to your router, and u cannot choose to have no internal as your router unable to talk to you. So u don't have a choice, your data will use internal ip(LAN) until your gateway which also means your router, and your data will be using public ip to get out from the router, when responded by whatever server, they will use your public ip back to your router, and your router will use private ip(LAN) back to you.

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Added on June 14, 2012, 11:48 pmTo know your private or LAN ip type ipconfig in command prompt. To know your public ip just access www.whatismyip.com. And your 1st post is wrong. When u are have Internet connection means u got public ip d. And when the u got a router, that means u got a LAN. Missing any, u can't access Internet.

im not sure what is minecraft server, but most servers out there, they use static or fixed ip, that is the most convenience or the easiest way, ppl know your ip and will never wrongly connect to somebody else.

and yes about no-ip, basically u got random ip, and u post your ramdom ip to a server with a fixed ip, and bind your new ip to something, ie UserID 2265 is always you, when u get a new ip, u tell no-ip.com which is using a fixed ip, and u tell them im user 2265, i have a new public ip now, which is .... and no-ip will record that, whoever wants to connect to you, will go no-ip and ask, where is User 2265 now, no-ip will tell them your new public ip. then others will know your latest public ip, they dont have to know your private ip, your router need to know, thats why its private. so connection will be made directly to your router, to certain port, and u should already configured when connection comes to whatever port, goto whatever internal ip(or lan ip or private, whatever term u like).

log in your router by using its ip, u will notice it has both internal and external, we call something like this a gateway, others cannot directly go to your pc without going through this. and u can log in your router with more than just 1 ip, if you allows admin login from WAN(or public), normally default only allow LAN.

the reason why ppl call WAN ip as public, is this number is unique in the internet, u cant have 2 same public ip. and private ip is private business, u can have your ip, i can have mine, they can be the same, as long as each pc ip in my network is unique, all works fine.

+1 for narf03

Other people mostly only say "get public IP lah... easier to torrent lah... ape lah" but never really know what they're talking about