|Do you mean incoming or outgoing connections.|
With outgoing connections it is straight forward, It is the modem/router that does all the work. For incoming connection you use port forwarding in the router to forwrd incoming requests to a particular computer which must be configure to accept it.
Computer dont keep a record of IP addresses it has used unless you tell it to.
If you are asking if the ISP know which particluar computer is connected then for the most part, they cant but there are circumstances where they can tell. Depends on what you are doing. Web masters can tell via their log on scripts.