A slogan or mark, for example, could be registered for use in movies but not computer products. The situation has changes recently but let us explain the difference down the page a bit. Originally, all goods and services were broken down into 42 classes. There is some bleeding between categories, and trademark examiners are unlikely to grant requests for nearly identical trademarks in similar categories, but class plays a role in granting trademarks. Recently it became necessary to list specifically the products or services to be covered, and the 42 classes have been expanded to a collection of specific sub-classes, which is reminiscent of patent classification, but far less useful.
Class is important as trademarks are class-specific.
One difficulty with trademark searches is that all the tools apply best to words which appear in trademarks. The solution appears to be image descriptors. I am uncertain of the international nature of image descriptors, but at least in Australia, there is a standard set of image descriptors. IP Australia allows you to search for other trademarks with a particular picture element - irrespective of the words involved. But to do this, you must first select the appropriate image descriptor.