I'm sharing the internet with someone. I have an Ethernet cable running into my computer from their router.

I used to have an Internet connection when I booted up. Now I have to use every time, after I boot up, to connect
This happened after I installed Slackware 12.1.
but after that the person I'm sharing with changed internet providers, and still have this problem.

Is there a way to fix this?
Would I need to do something at my end or is there something that needs to be done with the connection at the router? (maybe installing an operating system changed something with the connection to the router).

I admit that I know nothing about Slackware, but if there's a command you need to be run at boot time that your distro isn't doing for you (there's probably a way to make it do this, if you can be bothered to find out), then the quick and dirty solution is to put the command into your rc.local file.
For my distro, this can be found at /etc/rc.local
YMMV

Commands listed in this file will be run as the root user after the system considers itself to be "up & running". So it is a good place to add any "tweaks" or "fixes" you need.

Edit that file, as root, and add the command you need on the line before the final exit 0. Save the file, and reboot.

Should be fixed. Then you can spend some time working out why Slackware isn't doing whatever you needed to be done, for you.

did you try running netconfig to setup your network

Thanks.

I put dhclient in the /etc/rc.d/rc.local file.
I now get a connection when I boot up.

I did try running netconfig. That didn't fix the problem.