There are plenty of camera's with an ethernet connection out there.
They are very common for CCTV purposes.
They can send the video as a TCP network stream and usually have a web-based configuration/control interface on-board. Some also allow motion-jpeg caputuring or individual images as jpeg images. There is also a standarized software API (called ONVIF) that is supported by almost all camera's in the mid to end high ranges.
Biggest brandnames in the business are Axis, Bosch, Panasonic and Sony.
PS. Such camera's are not really designed for a point to point connection with a PC, but it will work.
Typical setup are multiple camera's connected by a normal computer network. Somewhere in the network is a control computer with a big harddisk. From here the camera's are controlled (zoom, pan, tilt) and the video-streams are stored. It's also possible that the control PC and the storage are separate (a NAS is often used for storage).
Disclaimer: I'm not directly involved myself, but I happen to work as an IT guy on a Bosch site where they develop such camera's.