A router can be used for several computers, whereas a modem is for one.
Therefore, if you look at a router you will four or more identical sockets numbered 1, 2, 3, 4. There will either be a network cable between one of these sockets and the network socket on your computer, or the connection will be wireless.
On a modem you will only see one socket suitable for connecting your computer, or the cable might be permanently attached to the modem. It will either plug into the network socket or the USB socket of your computer.
However, another way to know is just to read what it says on the device - just look at the label and see if it says "modem" or "router". Or, if it says "Speedtouch", for example, then it's a modem. The Speedtouch 330 seems to be one of the most common modems given away by an Internet Service Provider.
Finally, it could be a "cable modem", in which case it probably says "Sky" on it. This would be the case if you get your broadband, phone and TV as one package from Sky.