1>As per the above definition, is a modem a Network interface?
No. Like @Keltari says, It connects to a network interface on your computer. It also has a network interface to connect to your ISP's network, the telephone network, or otherwise upstream.
2> I read that a Network interface can have multiple ip addresses.
Does that mean that the modem at my home has multiple ip addresses?(I
have a regular modem that connects me to the internet)
Typically modems act as bridges, meaning they take what is coming out of the ISP's side and send it to whatever is connected to your side, and vice versa. Bridges don't have IP addresses - not in the way you are thinking.
The one exception is that the modem may be running an internal webserver where you can look up diagnostic info on your side. This is a separate thing from the internet service provided via this modem.
3>Is a router same as the modem?
Typically modems act as bridges, meaning they take what is coming out of the ISP's side and send it to whatever is connected to your side, and vice versa. If you need routing for the machines in your home network, a separate router needs to be provided.
Why would you need routing? So that multiple machines in your home can use the ISP's network connection as a gateway to the rest of the Internet, and vice versa.
Some ISP's give you a device that has both of these functions built in (some DSL "modems" are a combination of "modem" and router) - but they are separate functions. Usually on these you can disable the routing and tell the device to operate in "bridge mode" if you want to use your own router.
Although a 56k modem or a internal cable modem would be a network interface...? – Austin T French – 2013-05-13T02:03:40.763
What is generally referred to as a modem today is a stand alone gateway device running it's own operating system and which does typically have two or more network interfaces itself. The actual modem device (whether in a computer or gateway device) is a network interface. Based on context, the OP was referring to a gateway device. – YLearn – 2013-05-13T02:13:49.583
@YLearn But a "Gateway" is going to be a Modem / router combination, at least sometimes. – Austin T French – 2013-05-13T05:04:23.457
@AthomSfere A gateway has nothing to do with a router, it has to do with routing. You can set the gateway to be the same IP address and interface as the the device. – Keltari – 2013-05-13T05:33:22.570
@YLearn I am talking more about what devices are or can be sold as, as modem / routers are certainly sold as gateways http://www.bhphotovideo.com/bnh/controller/home?O=&sku=845331&is=REG&Q=&A=details Not that it is technically correct its useful information
– Austin T French – 2013-05-13T11:35:53.290