Router is a high RPM, low torque tool which can do magic with wood - make grooves, shape edges and a lot of other creative things. You cannot connect your Wi-Fi devices to this router. :)
Trim Router or Palm router is a hand held version of the router, which can hold 6 mm bits. Plunge Router is a heavier version, with more powerful motor, ability to change depth when the motor is on and which can hold 6, 8, 12 mm bits.
You would need a flush trim bit. This bit has a bearing at its end and a cutting edge, in line with the bearing. Bearing keeps cutting edge in line with the target surface so that you get a clean cut edge. Move the router bit quickly to avoid burn marks on the finished edge.
Use a simple technique to get fine finish when you are assembling 2 sheets of plywood. Leave an overhang of about 0.5 mm on the supporting side of the join. Trim the overhang with an edge trimming bit in a palm router. This will give a very fine quality join. Light sand if required.
For edge banding of teak wood edges on plywood/blockboard when building furniture, the same bit can be used to trim the overhanging teak beading. You can clamp the boards to your bench and move the trim router with your hand. When doing this, make sure to hold the router base firm on the edge, since more than half the router body will be floating in air.
You can also build a edge trimming machine as in this video, if you plan to make a lot of doors. For larger pieces, it is more convenient to use the trim router in your hand, instead of mounting it like in the video.
In addition to these applications, the bit can also be used to trim laminate after it has been applied on the target surface and the glue has dried.
Best value for money router is the Dongcheng DMP-02. It is available in the city market for around Rs.1500/- and online here.