Ok, let me try. (translation is off the top of my head, so check with the oxford translation).
Ὁμώνυμα λέγεται ὧν ὄνομα μόνον κοινόν,
"Things are said to by `homonyms' which have one name in common, . . ."
ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος,
" but the account of its being corresponding to the name (kata tounoma) is different . . . "
οἷον ζῷον ὅ τε ἄνθρωπος καὶ τὸ γεγραμμένον τούτων γὰρ ὄνομα μόνον κοινόν,
"just as a living thing and a drawing of one have one name in common . . ."
ὁ δὲ κατὰ τοὔνομα λόγος τῆς οὐσίας ἕτερος ἐὰν γὰρ ἀποδιδῷ τις τί ἐστιν αὐτῶν ἑκατέρῳ τὸ ζῴῳ εἶναι, ἴδιον ἑκατέρου λόγον ἀποδώσει.
" because the account of being corresponding to the name differs, . . ." the rest of the clause gets confusing after that though.
I'd say that what Aristotle means by λόγος τῆς οὐσίας has pretty well got to be what a contemporary philosopher would call a real definition. On Aristotle's view, we get definitions by empirical inquiry--gathering things together into genera and species in terms of their likenesses and differences, and then a definition (horismos or logos) which expresses the essence of the thing is just a summary of these genera and the specific differences, e.g. ``Man is a rational animal.''
His point in this paragraph is that mere linguistic use can be deceptive. Sometimes we use the same sound to name two things that in themselves have radically different natures. Aristotle's example is a man and a picture of a man, which apparently could both be called an 'anthropos' in Greek. A better English example might be a 'bank', which could be either the side of a river, or a financial institution. There is nothing that riversides and financial institutions have in common, so the use of the word "bank" is ambiguous in just this way.
The importance of Aristotle's point here will only really become apparent in the Analytics, when he is describing the kind of logical unity required for scientific inquiry. There he lays down the condition that you can't transfer scientifically proven conclusions about the nature of humans to pictures of humans. This is Aristotle's way of pointing out the fallacy of equivocation.
There is one more wrinkle, that is a famous issue in Aristotle's metaphysics and that is that Aristotle recognizes a special subset of homonyms where there is enough relation between the different senses to allow us to investigate them all under one science. He calls this "pros hen equivocity" although it's sometimes known in the literature as "focal meaning" or in older medieval texts as "the doctrine of analogy". This is how Aristotle is able to reconcile his claim that "being" is not a univocal notion--chairs and redness don't "exist" in the same sense, with his claim that metaphysics is the science of being qua being.