IOS is the operating system of cisco routers and switches. The name is a acronym for Internetworking Operating System. Usually you access it by telnet.  It is unix based operating system and everyone that know unix feels at home when telnetting into cisco router. Good resourse about the cisco IOS commands is:

http://www.cisco.com/univercd/cc/td/doc/product/software/ios120/12cgcr/rbkixol.htm

The trace command is used to visually see a network packet being sent and received and the amount of hops required for that packet to get to its destination.

Trace was created 1987 by Van Jacobson from a suggestion by Steve Deering, with particularly cogent suggestions or fixes from C. Philip Wood, Tim Seaver and Ken Adelman. It was originally created for unix and is called traceroute there.  In microsoft it is called tracert and in cisco ios it is called trace.

Good webpage about tracert in microsoft windows is

http://support.microsoft.com/kb/162326

And for cisco ios

http://www.cisco.com/en/US/products/sw/iosswrel/ps1831/products_tech_note09186a00800a6057.shtml

The ping commmand in cisco ios is divited into simple and extended ping. 

With simple ping you can ping one ip address. With extended ping you can use the ping for measuring many things other than simple conectivity between ip adresses. For example you can change the packet size.  A good webpage deschribing how extended ping works and the possibilities that you can use is

http://www.cisco.com/en/US/tech/tk365/technologies_tech_note09186a0080093f22.shtml

People can use private IP addresses inside companies or inside your home. They are not published to the internet. They are used for private use.

Public IP addresses are used for public use and they are published to the internet.  For example the public IP address of www.yahoo.com is 69.147.76.15