The significance of vases is roughly analogous to any other work of decorative art. "Decorative art" is a tricky term to pin down, but we might contrast it with the fine arts in this way: the fine arts are objects that are created entirely or almost entirely for aesthetic reasons. In others words, they have no use value beyond what they look like. Decorative arts, meanwhile have both aesthetic and use values. In the case of vases, this is perhaps obvious: a vase can both hold things (flowers, pennies, etc) and be beautiful or otherwise striking to look at.

Beyond that base-level significance, it is hard to generalize too much about "vases," which is itself a tricky term. For example, we refer to the ancient Greek practice of pottery painting as "vase painting," despite the fact that the objects in question were usually intended for holding various liquids (most prominently oil and wine, but also others), and had various unique names depending on purpose. Some of these names are calyx, krater, amphora, and lekythos. The differences between them might seem arbitrary to our eyes, but think of the differences between a pitcher, bowl, pint glass, and shot glass. This approaches the way that ancient Greeks thought about their different "vases" - specialized objects for specialized uses.

And, of course, ancient Greek ceramics are only one part of a vast array of pottery, metal, stone, and other "vases." In various other traditions these objects have various purposes, and their decorative motifs have various significances. In more modern times, vases and other objects of decorative art can be seen as primarily industrial objects, or they can be conceived of as "crafts," that is objects that seek to hearken back to earlier, more handmade traditions.

One generalization that we might make, though it might seem to be almost a tautology, is that the ubiquity of "vases" throughout human cultural production speaks to the ubiquity of the impulse they represent, i.e. to store liquids and finely-grained solids while also creating a visually-pleasing object that also has exchange-value in its own right.

I hope that has been a helpful overview - if you have questions about individual types or styles of vases (or periods of vase manufacture) I will be happy to do my best to answer them, or perhaps another flair may do so.