It depends on your set up on the network. If its just a home computer then you are assigned just one IP address. Although with my network we have around 6 different computers. So we got a switch to make fake IP addresses with in the home while only using one. For you, you should only have 1 real IP but maybe multiple fake ones. ex. lets say my IP is 172.16.0.2 but in my house its 200.100.50.2 This could be how you got mistaken.

Yes no maybe... I'm only going to talk about home computers here. Commercial setups are an entirely different matter. First - there are not enough IP addresses to go around, so it isn't practical to give everyone their own permanent IP address. This is being addressed with IP6 (?) MODEM users - every time you dial in, you are allocated an IP address for the duration of that session. When you log out it is released and given to someone else. Cable/DSL - You have a semi permanent IP address allocated when your modem is powered up. This is the IP address used by a single PC attached to the modem. This is allocated for limited time ('leased') and then released and renewed. There is a good chance you'll get the same one back since it's just been made available and you will probably be first in line. Lease time may be anything from hours to weeks. Home Network - The modem works as above, but this time the IP address is used by the router. The router then creates a sub-network isolated from the rest of the IP world. This network is usually uses IP addresses 192.168.1.x, and the router allocates an IP address (like 192.168.1.6) to each PC/print server/etc as it starts up. This is again a 'leased' address and periodically gets released and renewed. The exact IP address allocated to each device on a home network depends on hte order in which they power up. When your networked PC accesses the internet, the router receives the message, and then repackages it before sending it out with its own IP address. "Out there," all internet activity from all PCs on your network is seen as originating from your router, using the ISP allocated IP address. When replies are received by the router, it can tell which of the networked PCs made the original request, and it then repackages the response and sends it to (say) 192.168.1.6. For nearly everyone, this works perfectly well. There are cases where a static (unchanging) IP address is desirable or essential, like web servers and other specific computers that need to be accessed from out there on the internet, but this (a) costs money and (b) isn't usually necessary for home use

It depends on your ISP and the package you bought. You can often opt for a static IP address for a small additional free. When an IP address varies it does so because you are using DHCP.

It does depend on your ISP, but you can opt for a new IP address whenever you like, to speed up your internet connection.

Most internet users get a new IP everytime the modem is reset. It's what is called a dynamic I.P. It is allocated on demand. It's done for two reasons. One, limited number of I.P. Addresses in the world, at some point, there may not be enough available for everyone to have a dedicated address. Reason number two. It's more maintenance intensive to maintain a list of dedicated I.P. Addresses. Internet providers generally provide "Static" I.P. Addresses for an extra charge. That takes extra steps to attach a static address to a specific account or computer. It has some advantages when you run a server with a high volume of traffic. You don't have to pay a DNS listing service to track the server when the address changes. It takes less time to access web sites and other resources on a static address, because the internet doesn't actually have to search for the right I.P. address when you type "www.mywebpage.com" into your browser, it just goes straight to the last known reference. Some software applications require a server with a static address. By and large though, these days, 80 to 90% of everyone has a dynamic address.

in addition to the previous responses, if you are using a dial-up server, you will have a different address every time you get online.

It looks like all aspects of your question have been answered except for "why it varies in the latter case". If you are assigned a different dynamic IP every time you connect, it is (often) because the connection provider owns a limited "pool" (number) of IP addresses and doesn't have enough to allocate one to every customer. By "recycling" them, they maintain a functional network.

I live in a Latin American country, my case is somewhat peculiar. I have five numbers, one is my IP-ID between my server and myself. The second is my outgoing IP number, the third is my incoming IP number, the fourth is my domestic net IP number and the fifth is my public IP number. This last number changes as many times as I turn my PC off, the International Server I use has over a thousand numbers so I never know nor do I check on what IP I am actually using. The public IP changes each time I turn the PC on as it takes the available IP it can find, during one session it can change again many times if the IP number is saturated. Regards.