The origins of Roman money can be seen in the language. "One of the Latin words for money is PECUNIA, which originates from the shorter Latin word, PECUS, meaning 'cattle.' It may not at once be recognized that there is a connection between cattle and money, but a brief analysis will show a definite wedding of the two words." Source: Handbook of Ancient Greek & Roman Coins, by Zander H. Klawans, p. 177.
According to the Online Etymology Dictionary, "livestock was the measure of wealth in the ancient world, and Rome, like any other culture, was essentially a farmer's community. That pecunia was literally 'wealth in cattle' was still apparent to Cicero."
To the early Romans, cattle were a unit of exchange. "In early times, cattle were viewed as a trading commodity (as they still are in some parts of the world), and property was often valued in terms of cattle." -- Merriam Webster.
This arrangement worked well in the early Roman world. "Rome under the kings and in the early republic managed without a coinage," according to the Oxford Classical Dictionary (3rd ed. revised), but the system broke down when the ancient economy got larger and started to expand to more distant markets. Coins were invented because cattle were too big to be practical for business transactions:
"Obviously, such bulky material was an inadequate means of exchange," according to the Handbook of Ancient Greek & Roman Coins (p. 177). "The problems of transporting hundreds of heads of cattle here and there was a great one and it became more intense when the world markets of those days started to blossom; trading in cattle became cumbersomely impossible. The logical answer was the creation of a means of exchange, universally recognized, but small in size. Coinage was the answer."
I don't think it's correct to say that these early coins were a "cattle-backed" currency in the sense that one coin represented a certain number of cattle. The coins had value in and of themselves because their value was directly related to their weight (in bronze, for instance). In fact, the first Roman coinage "was crude, shapeless and heavy in an attempt to approximate the value of the coin with its actual weight." (Ibid).
Some of these coins "carried the images of animals, such as birds and cattle upon them, and also inanimate objects, such as tridents and shields. It would seem to be a mistake to call such pieces "money" rather than to consider them as having been used as a means of exchange on the basis of weight." (Ibid).
The difference between "money" and "a means of exchange" escapes me. According to Wikipedia, at least, money is a "medium of exchange." Whatever the case, the value of these early Roman coins appears to have been based on their metal value (otherwise their weight wouldn't have been an issue). The value must have been purely relative, i.e., the larger the coin or the higher its actual metal content, the more it was worth.
The switch from a system of livestock-based commodity exchange to metal coinage was an important factor in the expansion of Rome, but it introduced other problems. Despite the current myths about the "intrinsic value" of gold, silver, etc., metal coins can be debased and inflated like any other kind of fiat currency. This can be done by "clipping" the coins, i.e., by using various tricks to reduce their actual metal content. In later years, Roman emperors did just that in order to cope with their money problems. The inflation of the Roman currency was one of the factors that led to the fall of the empire in the West.