The sources on Roman music are much fewer than those on ancient Greek music, despite a great deal of the latter being older. We have Greek musical treatises as well as scores in a comprehensible musical notation system, whereas virtually no such documents have survived, remarkably, from the entirety of Roman history^1. Ulrich and Pisk, in A History of Music and Musical Style, claim that the artform as a whole had a lot less importance to Roman than to Greek culture, and indeed the Latin musical treatises we do possess (such as Boethius's De Institutione Musica) are almost exclusively concerned with contemporary Greek and Hellenistic music. It wouldn't be surprising if Roman music, like its Greek model, was essentially monodic, i.e. had a single melodic line with no harmony or counterpoint, and possibly developed as a way to accompany epic or lyrical poetry.

Most of the evidence regarding Roman music comes from the iconography, alongside a few surviving instruments and mentions in literature. For instance, we know from the Satyricon that the wealthy could afford singing slaves (although Petronius, in typical fashion, presents it more as inconvenience than entertainment). From Roman imagery, we know that music was part of religious celebrations -- most notably the Bacchanalia and other syncretic cults inspired by Greece from the second century BC onward --, military pomp, theatre (the most distinctive form being the pantomime, where a dancer accompanied by a singer and sometimes a flautist using a scabellum acted out a story) and, later, personal entertainment. Roman instruments that have survived either through artefacts or representations include the organ (including the originally Greek hydraulis, a rather nifty water-powered mechanical organ), the tibia, a reed or flute also of Greek origin, as well as the Roman tuba and the bucina (a kind of ancient trombone), both likely of Etruscan origin, which would make them two of the few natively Latian instruments. (It should be noted that the latter two are almost always depicted in military contexts.) The Romans had also a number of stringed instruments, most of them of Middle-Eastern or Greek origin, like the sambuca (a sort of harp), the lyre and the cithara, and percussions like the tympanum.

Because of the paucity of sources regarding the actual sound of Roman music (and, again, the complete lack of scores), we have little information on the melodies or tempos that were common at different periods of Roman history, but we do know which instruments were commonly paired: as mentioned, the tibia and scabellum (a percussion similar in operation to modern castanets, but operated with the foot) were used together in pantomimes; ancient depictions of Roman "bands" show rhythm sections of cymbal and tympanum players accompanying harpists, flautists or organists, etc.

(1) The single surviving fragment of Roman music was long thought to be a musical phrase accompanying a line of a play by Terence, the Hecyra, copied by an Italian composer in the 18^th century. Its authenticity is now considered to be pretty doubtful, as far as I know. In case you're curious, you can hear a modern rendition here.

Very late edit: There has been a very good answer to that question in this sub already by /u/racecar_ray, it turns out! https://www.reddit.com/r/AskHistorians/comments/64e3ar/what_did_roman_music_sound_like_and_what_form_did/dg1v1uc/