What is cold-pressed honey?
This is most likely referring to honey processed without any heating, often simply called raw honey, or possibly cold-extracted or cold-processed. It likely has more flavor and color preserved than commercially-processed honey (which uses plenty of heat), but it may not be much different from gently heated honey. It's not actually cold (honey flows really slowly when cold), just not heated.

There are two primary ways honey would have been heated during processing. First and most common, most commercially sold honey has probably been heated substantially (perhaps up to 165F - 73,89ºC) to help melt crystals, allow impurities to separate, and to make sure everything's dead and safe. This definitely does affect the honey - it makes the flavor a bit more mild, and the color a bit lighter.

If you're getting honey from a very small producer (maybe someone with a hive in their yard), it's also possible that they actually heated it in order to separate the honey from the beeswax. The most effective way to get honey out of the comb is to remove the caps and use a centrifuge to spin the honey out of the frames, but for hobbyist beekeepers without access to an extractor who don't want to just crush and strain it, another method is to heat the comb, letting the melted wax (it melts around 145F - 62,78ºC) float the top for easy separation. This is pretty gentle heating, and doesn't have nearly as strong an effect as the commercial processes.