Why does most routers has default ip gateway 192.168.1.1? How do I change the gateway? If it gets changed, does every pc connected get a modified ip?
There are various non-routable (private) address ranges. See https://www.rfc-editor.org/rfc/rfc1918 and http://en.wikipedia.org/wiki/Private_network for details. The various ranges allow for varying numbers of subnets and hosts within a subnet. The reason 192.168.1.1 is so common is that it falls within the smallest private address space (192.168.0.0 - 192.168.255.255)... home networks, relative to business networks, are very typically small so the larger private address spaces (172.16.0.0 - 172.31.255.255 and 10.0.0.0 - 10.255.255.255).
It's not possible to provide specific assistance with changing the configuration of your router without knowing what make/model it is (and version of software it is running). I can simply suggest you to look into changing its "internal" interface IP address. The configuration of the interface/network will determine the clients' configurations.
For example, in a typical residential broadband connection, if you change your internal interface from 192.168.1.1 (subnet mask 255.255.255.0, network 192.168.1.0, broadcast 192.168.1.255) to 10.0.0.1 (subnet mask 255.0.0.0, network 10.0.0.0, broadcast 10.255.255.255) the clients' gateways will be 10.0.0.1.
Your questions are actually very broad so I hope the above helps and gets you started searching further.
I believe is because it's easy for remembering - knowing that the router is part of the C network (if we speak for home or small business network), you can easily guess what is his default gateway - they are usually 192.168.0.1/1.1 or 2.1; and yes, you can change it by the router settings, under LAN settings menu.
like user1801810 said 192.168.xxx.xxx /24 (255.255.255.0) is a class C private address (this means that it can be used by anyone, on a private network such as a home or a office) this traditonally gave 254 usable address e.g. 192.168.0.1 to 192.168.0.254 where 192.168.0.0 was the network address (mainly requried for DCHP (Dynamic Contol Host Protocol)(what assigns your IP addresses)). And 192.168.0.255 is your broadcast address.
Where through the usage of VLSM (Variable length subnet masks), the class-full system is all but obsolete, the base addressing scheme e.g.
Name Address range Number of addresses Classful description Largest CIDR block 24-bit block 10.0.0.0–10.255.255.255 16777216 Single Class A 10.0.0.0/8 20-bit block 172.16.0.0–172.31.255.255 1048576 Contiguous range of 16 Class B blocks 172.16.0.0/12 16-bit block 192.168.0.0–192.168.255.255 65536 Contiguous range of >256 Class C blocks 192.168.0.0/16 (https://en.wikipedia.org/wiki/IPv4#Private_networks)
simple example (N = network section H = host section) NNN | HHH HHH HHH Legacy Class A 10.0.0.0 to 10.255.255.255 (address range) /8 (cidre = 255.0.0.0 - subnet mask = 0.255.255.255 - wild card (only used when configuring certain routing protocols e.g. OSPF and EIGRP)
NNN NNN | HHH HHH Legacy Class B 172.16.000.000 to 172.31.255.255 (address range) / 16 (cider = 255.255.0.0 - subnet mask = 0.0.255.255 - wild card mask)
NNN NNN NNN | HHH Legacy Class C 192.168.0.0 to 192.168.255.255 (address range) /24 (cidre = 255.255.255.0 - subnet mask = 0.0.0.255 wild card mask)
wikipedia has good articles on both subnet masks and wild card masks
These (address if not in totality due to VLSM (variable length subnet masks) are still used by network admins, to help predetermine the size of network which they are dealing with. I hope this answers question 1 of 3 which you asked.
as for question 2:
each router is different, there should be no reason you could not change what address range it distributes (as long as it co insides with the above range). however with out the actual router make and model I can not advice up on that.
as for question 3.
every pc typically will get their IP address from the router via protocol called DCHP, when your PC is connect to a network it will send a DHCP request using UDP, to ask for a IP address. If a DCHP server hears this request and has address left in its pool then it will temporally reserve an address and offer this address to the client in this case the PC. the DCHP address pool is configured upon what the addressing system has been set to. e.g. 192.168.0.0 /24 then it will release an address between the ranges of 192.168.0.1 to 192.168.0.254 (inclusive) however they may be (almost certainly) "reserved addresses" which are address which should not be sent via DCHP, these are typically the address of any routers (default gateway), manages switches and servers which are on the network.
Hope this helps
the default gateway address comes from the old class-full addressing system
it should be possible look in your router's setting
the address come from a protocol called DCHP