A lexical category is one particular kind of syntactic category. The terminology here quickly gets more complicated than helpful, but what defines a lexical category is that all the words in it are bound by the same restrictions on how they combine with other words.
Most of the big lexical categories are probably already familiar to you. Nouns refer to entities (cat, hat, beer, freedom, etc.), adjectives modify nouns (fat cat, blue hat, cold beer, individual freedom, etc.), and verbs position nouns in terms of a specific statement or claim (I kicked the fat cat, I bought the blue hat, I want cold beer, I have individual freedom, etc.)
Countless different lexical categories have been proposed, and different linguists count different things as lexical categories. But, the major categories and their descriptions should be easy enough to find if you look up (like Corybobory suggested) "parts of speech".
In figuring out how particular sentences are structured, the easiest thing to do is usually to find the top-level verb (which, in English at least, will generally be conjugated for tense) and then figure out how all the other words relate to it.
In your example sentence, the main verb is "owns", because that's what establishes the activity of the sentence. Then, you have "my brother" (who does the owning) and "two very fuzzy cats" (which get owned) which are noun phrases.
