A basic router function is to allow different connections to be made between different subnets or network segments. This is the basis of the Internet. Your address on the Internet will pass through many routers as packets carrying it traverse the Internet to reach a remote site. That site can use your Internet address to send any responses back to the originator. The responses do not necessarily come back along the same route as was taken by the outgoing packets, and can pass through routers that have no record or information about the packet to which a response is related. The originating IP address has to pass through all the routers as the packet crosses the Internet so that the response can be returned.
To conserve the usage of addresses on the Internet, a new class of router was devised for use in the home and in some organisations. These include a feature called Network Address Translation (NAT). With NAT, a single address on the Internet can be shared by multiple devices on a private network inside the home or organisation. The IPv4 specification reserves three ranges of IP address that are not allowed on the Internet itself, and are for use specifically within a private internal network. Each computer on the private has its own address, normally allocated from these ranges. As an outgoing packet passes through the router, the router translates the originating IP address to the single address assigned for use on the Internet. When a response comes back, the router recognises it and does the reverse translation to allow the response to be addressed to the correct computer on the private network. Unlike the routers on the Internet itself, where a response may come back along a different path and through different routers, with a NAT router, the response MUST come back through the same NAT router as the original outgoing packet used as no other router will not be aware of the mapping to translate the returning packet and direct it to the correct computer on the internal network.
A NAT router provides protection from unsolicited packets coming from the Internet, such as random attacks, as the router will not have any mapping and cannot forward any such packets, which will simply be discarded.
So, to answer the initial question, home routers normally change the computers IP address as it passes on to the Internet. A general router does NOT perform address changes.
I hope this helps.