What is the purpose of a splitter?
It sounds like you're interested in the practical reasons for this device, not just what it does electrically, so I'll answer for that perspective.

The reason there is a splitter in the kit is to allow you to connect both a TV and the provided cable modem — each requiring their own connection — to the cable coming into your house.


  My guess is there is one input (the top one) and two outputs (the lower two) for the splitter in the picture. Are the two outputs two signals that are identical to the input?


If you consider the top as an input, then the other two connectors are identical outputs. However, your cable modem transmits as well as receives, so there are signals traveling in both directions. The top should always be the one connected to the cable coming through your wall.


  In other words, is a splitter used like a switch/router for Ethernet network?


Not quite. It is not like an Ethernet switch, in that it is asymmetric: if you tried to communicate between the two bottom ports, you will see more signal loss than between either bottom port and the top port. With regard how you connect it to devices, the configurations are similar to a router (or switch with uplink port), but the actual electronic characteristics are completely different — for one thing, the splitter is a passive device — it does not interpret or modify the signal, only pass it on, ideally unchanged (but somewhat reduced).


  Do the cables connected to both input and outputs belong to the same type, such as belonging to coaxial cable for the splitter in the picture?


This sounds like you are asking about some more general class of devices. I'm not sure what I could say about that class, but there's something interesting to be said about this splitter: yes, you do use all the same cables, and for good reason: an important characteristic of cables and connectors carrying high-frequency (RF) signals is impedance. Typical cable-TV hookups have an impedance of 75 ohms. If there is a mismatch at any point, then some of the signal passing through that point is reflected, and is effectively lost, for purposes of getting it to its intended destination.

If you've ever connected an indoor antenna to a TV, then perhaps you've seen a "300 ohm to 75 ohm" adapter with screw terminals, which besides adapting the connector also converts between impedances — it is a matching transformer (as well as a balun, which is another topic entirely).

A practical concern is that if you do not have a TV (or other cable device) as well as a cable modem connected to your cable service, you should not put the splitter in your line, because an unconnected port on a splitter is an impedance mismatch (specifically, it has infinite impedance, also known as an open circuit) which will degrade the signal for the other device. You could also connect a terminator to an unused port, which is just an appropriate resistor (75 ohm, for TV) across the conductors. However, it's better to leave out the splitter, because a splitter necessarily introduces a certain loss of signal (in order to split it evenly between the two ports).

(Impedance matching is also why coaxial cables should not be crushed or bent sharply; changing the shape changes the impedance.)