This may be a stupid question showing my lack of understanding of taxonomy, but I was wondering if the current classification system we use contains broad enough categories to include new life forms, if they are discovered on other planets? Would this depend on whether or not the new organisms could be shown to be evolutionarily related (eg have DNA), or are the highest levels of organization independent of relatedness and more dependent on physical similarity?
There are basically three highest taxonomic levels: bacteria, archaea and eukaryotes. Many sources distinguish only prokaryotes and eukaryotes, subdividing prokaryotes into bacteria and archaea. Other sources posit archaea as a third domain on the highest level. That's a long debate though and fortunately not the topic here.
By appearance, the dual-domain system makes more sense because archaea and bacteria look very much alike under a microscope, and can both be clearly distinguished from eukaryotes (which have membrane bound nuclei). At the same time, comparing genetic structure and the protein machinery that interacts with DNA, eukaryotes and archaea are very similar and bacteria are clearly different from both of them. This demonstrates how appearance and molecular organisation have been typical criteria for distinction between branches of the evolutionary tree.
There are, however, many features that even those three most basic branches of life share, and they are in no way simple or few in number. For example, all extant life uses DNA to store genetic information and proteins for almost all processes governing the various reactions going on at any time in a cell. However, RNA for example is able to perform both the information storing function of DNA as well as the catalytic functions of protein. This just shows how the principles behind life are not unique to the molecules in use now, and extraterrestrial "life" may work by the same principles on an entirely different molecular basis.
If you want to know whether life found on another planet could be fit into our current classification system, the answer would be: No, most likely it will not be classified using the current domain/kingdom/phylum system. This is because our system describes evolutionary descendency, so unless this newly found life shares some common ancestor with an organism on earth, it would have to be classified in an entirely new system. (Of course this is subjective as you may find it more likely that it does share a common ancestor [Panspermia], while I personally find it more likely that life would have evolved entirely separated on this other planet seeing as we haven't heard of any life within reasonable distance.)
However, it is possible that this new life will follow the same evolutionary rules such as natural selection, and similar evolutionary pressures may drive to evolve similar properties. It may then be possible to draw parallels to characteristics of the current classification system, especially based on molecular function, and possibly appearance (micro- and macroscopically).