The other answer to this question does an excellent job of noting the varied ways the dragon is described and discussing these descriptions' significance. To build on these ideas, I'd like to focus specifically on the fact that the dragon is described as greedy, as this detail is quite important to the poem as a whole.
To begin with, the dragon is described as a greedy beast that lusts after treasure. For example, take a look at the following description:
[The dragon] is driven to hunt out
hoards under ground, to guard heathen gold
through age-long vigils, to little avail. (2275-77)
In this quote, we can see that the dragon greedily amasses wealth for no particular reason; he doesn't use it for any constructive purpose, but he still wants to have as much of it as possible. This greed offers an important contrast to the generosity of Hrothgar and Beowulf. Both men are generous kings, rewarding their warriors with riches in return for loyal service while also providing their subjects with peace and plenty. They are the opposite of the dragon, as they do not amass wealth, choosing instead to distribute it freely. As such, by describing the dragon as greedy, the poet also shows the destructive nature of greed and illustrates why it's important for a king to be generous.
The author (or authors) of Beowulf don't really devote a lot of effort to describing the dragon physically; for example, we don't know how large he is, or his exact shape. We do pick up a few details though.
We learn that it is at least three hundred years old, and that it is "cursed" to guard the treasures of the dead. It is given titles such as "plague-of-the-people" and "warden of gold", and words such as "snuffed" and "belched" describe some of its behavior, giving it an animalistic quality. To compliment and contrast this, it is also said to be "greedy", "boiling with wrath", and "war he desired", traits that we might ascribe to a more intelligent being. He is also said to be fearful of Beowulf when he realizes what a powerful foe he is facing. It is no secret that Tolkien took inspiration from the dragon for his depiction of Smaug in The Hobbit, and we can see many direct parallels between the two dragons.
Most prominently, the dragon is strongly and repeatedly associated with his ability to breathe fire; he is said to be "folded in flame" as he goes forth to wage destruction in revenge for the theft of his treasure, and it is also mentioned that the dragon can fly.