If your DDNS thing is working for someone trying to access your website hosted at home, then it's a public one. I presume (never used one) that a DDNS would not work on a private IP in the first place.

If you go to http://www.whatismyip.com/, it will tell you the IP address that the rest of the world sees you as.

If this address differs to the one your ADSL router (not your PC) is reporting, then you have a private IP address.
If it is the same as yours, you have a public one.

If, when you disconnect and re-connect your ADSL modem, your IP address changes, you are using a dynamic IP address, otherwise, you are on a fixed IP address. Note: this test is not 100% guaranteed, the ISP could re-assign you the same IP if you haven't been disconnected long.

What happens is that there aren't enough public IP addresses for everyone at the moment, so most ISPs have a "pool" of IP addresses that is shared between their customers. No computers can have the same public IP address at the same time, but on the premise that not every customer is connected 100% of the time, it is ok to share the "pool" and dish addresses on demand.

Fixed IP addresses on the other hand are allocated once and for all to a customer, and usually cost a little extra for the privilege of using a scarce resource.

Private IP addresses are plentiful however, as they belong to a number of ranges that MUST NEVER be used on the public internet. If you have an ADSL router which has several ethernet ports and dhcp, chances are you will be using IP address in your network that are like 192.168.0.x.

Contrary to public IP Addresses, several thousands of people are currently plugged into a network and have say 192.168.0.10 as an IP address.

If they have access to the internet, they will be "hidden" behind a router (adsl router for example) which has a single IP public address on the outside, or, an ISP allocated private IP address which itself is hidden behind yet another router which will have a public ip address.

pheww, hope this helps and doesn't confuse any further.

As said above, if you don't know why you need a public or fixed ip address, you probably don't need one