Example of a router is here:
Black & Decker Tools | 10 Amp Variable Speed Plunge Router RP250
What you showed is a grinder. It is not really meant for cutting, but people definitely use it to cut GI pipes, MS sheets, etc

because it can behave like a circular saw. A router is used to cut (mostly) pre-defined shapes (e.g. the decorative edge of a table, or bevelling the edge of any panel, etc).
Examples of various router bits/blades to cut specific profiles:
Handrail / Table Edge Router Bits by Amana Tool
Just for info, jig saw is a reciprocating type of saw (as against circular blade). Like this:
Black & Decker Tools | 5.0 Amp Variable Speed Orbital Jigsaw JS660
It is really nice to have these tools (and many others) if one has the room to get down and dirty.
I have some of these tools but using them is subject to very loud noise and the bothersome saw dust, so I would rather get my cutting and routing done at neighbourhood jig saw workshops.
Another nice "accessory" to have is round wood bits of 1/2, 3/4, 1, 1-1/4 inch. They do come in handy when doing any wood working involving making large holes.