Put the catalog file and all photos on an external hard drive, move this external HD back and forth to whatever computer you are working on.
The external drive solution that DJ describes is the best. I have also heard of some people having success with Dropbox, but that could require a lot space if you have a lot of images.
The bottom line is that Lightroom does not have a solution to 'sync' multiple catalogs, or even collections within multiple catalogs. The only way to be in sync is to have a single catalog that is accessed from multiple locations.
Lightroom is a single-user, single-computer software system.
The easiest (not prettiest) method of using it on multiple computers is to put the catalog and the images on an external drive and move the drive from computer to computer as needed, as already suggested.
You can often setup Dropbox.com or similar file sharing services to copy the catalog from computer to computer, but there are risks involved with jumping though these hoops.