When running ethernet cable over 100m will a router in between the two 100m segments do the same thing as a switch would, allowing me to go 100m, router, and then another 100m? Or am I going to have to purchase a switch or a repeater to do this?

First, I wouldn't go to a full 100 metre length. That's the maximum rating for Cat 5 cabling but in my experience, you don't want to exceed 85 metres or your signal (connection) will be flakey and problematic. A router should do the same thing depending on the router. Whatever you have to use, or are planning to use, read the specification sheet on the hardware to find out if it will.

A router, switch, or hub "rebroadcasts" the signal from the WAN port to the LAN ports so yes, you could get another 100m out of it. As the poster above said, you shouldn't go the full 100m between repeaters unless you have really high quality cable. You should really consider moving to fiber optic cable or even microwave relay if you're going to be joining really distant segments of a network. Not only can you get much longer segments with fiber optic you can boost the bandwidth up into the gigahertz region.

Thank you both for your speedy responses. For now I think I will just stick with the cat5e cable, just becuase it is cheaper, and the lengths aren't quite 100m, that was just more of a hypothetical. The line is only going to be used to check e-mail, so it doesn't have to have super high bandwidth. But thank you both again for helping me out with that question and helping me sort out my options.

And just to clarify the connection options, when using a router in this situation you would need to disable the DHCP function and connect to the LAN ports not the WAN port. This uses the router's switch function only. I used to have a signature but it disappeared and I just couldn't be bothered writing another so please feel free to ingore this.

You can add devices to increase distance. That is how the world wide web does it. See also devices from Patton and others that extend distances for copper or fiber.