Ideally, the reviews in Zentralblatt and MathReviews should provide precisely this information. In practice, many of the reviews are not exactly illuminating. The most important papers, however, often end up getting a "Featured Review" on MathSciNet, and those are usually very clear and well written.
Now, one other way of learning about the significance of papers (if either the paper itself does not provide sufficient introduction so Gerry's advice is hard to follow, or if for some reason you prefer to seek out third-party evaluation of the paper) is to follow the citation trail. Go to the MathSciNet entry for the corresponding paper. On the top right there is a box listing citations to that paper (these are reasonably complete for papers published within the last 40 years). If there are Review articles citing that paper, go and read it. You will likely be enlightened, since the articles will probably also give a good overview of the "lay of the land" so to speak. Citations from references are a bit more hit and miss: sometimes the paper is only cited for a minor technical fact, sometimes the paper is only cited to be polite, but sometimes you will find another paper in the field with a good expository account of why precisely the methods and results of the paper you are interested in is useful.