It won't be easy to define Cicero's style here. It's a huge topic. So see Michael von Albrecht, Cicero's Style: A Synopsis (Brill 2003). I can help you get access to that, if you are interested. Warning: there is a lot of Latin.
In basic terms, Cicero employed a richness (Latin abundantia) of language but a precise word choice while adhering (more or less) to a clarity and balance of structure in thought, sentence, paragraph, and composition. This is incredibly vague because Cicero's style obviously evolved over his long life, and also varied among the many genres he produced: political speeches, courtroom advocacy speeches, private letters, philosophical works, even poetry.
One of the reasons why Cicero was so valued and emulated later, especially in the Renaissance, was the simple fact that more text of Cicero survived than any other. For the 1st century BCE, Cicero's writings dominate the available evidence. We have Caesar, of course, but much less, and Caesar and Cicero were writing in different spheres and to different purposes. We have a bit of Varro, we have the anonymous Rhetorica ad Herennium, and, at the very end of the century, Livy, who was also writing in quite a different style and for quite a different purpose. Cicero is the example par excellence of Republican Roman rhetorical style. His finicky choices of vocabulary become "canon," and his tendency to use more abstract nouns than had been common in earlier periods becomes canon. And so forth. We get glimpses of other important masters of rhetoric in Cicero's surviving dossiers of letters. A letter to him from Sulpicius Rufus, concerning the death of young Tullia, is often considered to be the best extant example of Latin prose. Sulpicius Rufus in this letter and Cicero (when he's really trying) are very comparable. One gets the sense that what we call "Ciceronian" was in fact probably "well-educated Late Republican Roman." Other times, we get more strained, less sophisticated examples of prose, like the several letters from Marcus Caelius Rufus, Cicero's old pupil. Marcus Antonius sends Cicero a menacing letter which is clearly meant to be a rhetorical masterpiece, but just is not. His son-in-law Dolabella also sends a long, stinky letter to him.
Some common elements of Ciceronian style:
We shall go on to the end, we shall fight in France, we shall fight on the seas and oceans, we shall fight with growing confidence and growing strength in the air, we shall defend our Island, whatever the cost may be, we shall fight on the beaches, we shall fight on the landing grounds, we shall fight in the fields and in the streets, we shall fight in the hills; we shall never surrender...
exclamation (O tempora! O mores!)
rhetorical questions
periodic arrangement and long, complicated but balanced sentences in general, often incorporating tricolon, the use of groupings of three, often in ascending order of impact (or the opposite)
There are many more examples. These were common rhetorical devices which Cicero had learned well and employed to good effect. None of them are unique to Cicero, and in fact Cicero owed very much of his style to Greek orators and philosophers of centuries earlier like Demosthenes and Plato, whom he knew well and always sought to emulate. Why Cicero, instead of Plato, became the object of emulation for the Medieval period is a complicated question related to availability of texts, knowledge of Latin and Greek, and perceptions about Roman vs Athenian society.