A modem usually refers to the device in your house that receives the internet signal. This can be both broadband, fiber optic, telephone line or just anything else really that is connected to the internet.
The modem then, through an ethernet cable also known as RJ45, sends the signal to the router.
The router is the device that administrates your network. It gives each PC a unique (internal) IP for example. Some years ago, you'd usually link the router up with a switch, to which you connect all the PCs on the network. Nowadays, the ISP usually gives a router with builtin switch, or a router with a builtin wireless access point.

What you've got, is apparently a modem-router-wireless access point. I'm guessing there's also an ethernet port in this device, which you can plug other devices like switches or wireless access points into. This means, that you don't really need the thing that you linked, you just need an ordinary wireless access point.

I suggest something from Linksys. With a budget of around $80, you'll get a fairly good access point with good range and stability.