Catullus arrived in Rome just in time to join a literary movement we normally call the neoterics, or "new poets." These were young, financially independent urbanites very similar to the les Fauves artists of the Salon d'Automne (or any such Parisian art movement). Catullus was rich, but a relative outsider to Roman high society: he was from Verona, but was highly-educated and (apparently) well-connected to the capital. His fellow neoteric C. Licinius Calvus may have been his introduction to the City: Calvus was a junior orator at the time, son of a praetor, and with consular ancestors. It is possible the two met while studying abroad.

The Roman neoterics were intensely focused on Greek poetry, especially the works of 3rd-century Alexandria (who themselves were called νεωτερικοί, "new poets"). The Roman neoterics, like their Greek predecessors, rejected traditional values and the traditional literary bounds that came with those values. Catullus' style is playful, trans-genre, but meticulously crafted. Callimachus of Alexandria was the neoteric hero. We also find many elements of Archaic Greek poetry (like that of Sappho), but the neoterics were distinct in their blending of Archaic and Alexandrian Greek poetic forms with "native" Italic forms. I say distinct, but this is probably misleading. What is preserved from antiquity is often mere accident, and we have the poems of Catullus from a single early manuscript ("Manuscript V, of Verona"). Without this mysterious and now-lost text, we would have only a few Catullus poems. We have virtually nothing from the other neoterics--only their names and some scattered references and fragments in passing. That includes Licinius Calvus, Gaius Helvius Cinna, and I might suggest that Catullus' famous lover Clodia penned a neoteric poem or two. A common feature of Latin neoteric poetry is a dialogue (now lost) between different participants, always in poetry and often with playful innovations on meter or genre. Having only Catullus and a few fragments severely limits our understanding of this inter-poet interplay. Publishing the poems as books (Catullus calls his a "little book") seems (to me) to have been an afterthought.

We call them the "new poets," but they surely did not spring up in Rome like mushrooms. Like les Fauves, they were likely the newest artistic movement in a long sequence of such. None of the earlier examples of Roman poets of this type survive, though we have circumstantial evidence that young Romans had been admiring and imitating Alexandrian output for decades or even centuries. In 85 BCE, a young Cicero famously adapted a Latin translation of Phaenomena by the Alexandrian Aratus.

The other three you mention, Tibullus, Propertius, and Ovid, all come slightly later than the heyday of the neoterics. Sometimes the three together, along with some others, are called the Roman love elegists or "Augustan" elegaic poets, but this is a bit of a misnomer. Unlike the neoterics, all of these had wealthy patrons, though those patrons were not necessarily directly connected to Augustus or the machine of Augustan poetic propaganda. Tibullus never mentions Augustus directly or indirectly. Their elegaic style owes a lot to the neoterics, but is in general more restrained, and less obtuse in vague Hellenistic references. A lot of the comparisons are highly technical (regarding vocabulary choice, word position, metrics, etc) and I won't get into it here. The go-to article would be Paul Allen Miller, "Catullus and Roman Love Elegy," in the Companion to Catullus volume.

Roman love poetry continued in the early Empire, though we do not have very much extant from that period. It probably changed quite a bit, along with the rest of Roman society, but by the time we get to the poet Martial, we find a reverence for and imitation of a lot of the core tenets of the neoterics. It is very clear that Martial, and his literary peers, both had access to neoteric poetry and knew it well.

I mentioned that most of these post-neoteric poets had patrons, and that highlights what was perhaps an important shift in Roman society at the time. There is a book by Stroup that dives into this topic in incredible detail. It's a bit hard to read and understand, but here's the reference: Stroup 2010, Catullus, Cicero, and a Society of Patrons. Cambridge.

As mentioned, an excellent place to read more about all this is the Catullus entry in the excellent "Companion to..." series by Blackwell. This one is edited by Marilyn Skinner (2007).