Each Latin noun does indeed have a particular intrinsic gender, which needs to be learnt when the noun is learnt (although with a very little experience it can usually be guessed correctly).
A Latin adjective, in contrast, as a rule alters its gender to agree with that of the noun or pronoun modified by the adjective. So a masculine noun takes a masculine adjective.
In Latin grammatical gender is less closely tied to biological gender than it is in English. It is true that for the most part Latin nouns denoting male persons and animals are masculine, while those denoting female persons and animals are usually feminine. But nouns denoting things with no biologic gender may in Latin be masculine, feminine or neuter. Femina (woman), for example, is feminine, while vir (man) is masculine, but ager (field) is also masculine, urbs (city) is feminine, and mare (sea) is neuter.
A few Latin nouns may be masculine or feminine, depending on the biological gender of the person or animal denoted. Examples are accola (neighbor) and pavo (peafowl). Such nouns may be listed in dictionaries as m or f, or as c (for common). I prefer to use the former for words for which there is uncertainty about the proper gender (see further a little below), and "c" for words like pavo and accola. These latter may be called epicene, or in Latin [verba] utriqui generi communia, that is , words common to the two genders. I usually just say communis (with whatever ending is called for) or common. In Latin as in English we may if we wish specify biological gender by adjectivally modifying a noun of common gender. Pavo mas or pavo masculinus means male peafowl, that is peacock, while pavo feminina means female peafowl, that is peahen.
Some Latin nouns exist in two closely related forms, one masculine and one feminine. Equus, for example, means horse, either a stallion or a horse of unspecified sex while equa means mare. Magister is a male master or teacher, while magistra is the feminine equivalent. Textor is a weaver if male or of unspecified sex, while textrix is a female weaver. Such pairs are sometimes called substantiva mobilia-- a term which I dislike.
A great many nouns denoting persons which are traditionally masculine, and a few which are traditionally feminine, are perhaps better today regarded as common. Nauta, sailor, for instance, is traditionally masculine; but if the sailor in question were female, one might well treat the word as feminine” in other words, I think that nauta has become common with the changing times.
A fair number of Latin nouns may be said to be of uncertain gender. They have historically been regarded as of two different genders, by different respected authors, or even by the same author at different times. Such words are said to be heterogeneous. They constitute one example of redundant nouns, in that they possess one (or even two) genders more than necessary.
It is also common for a Latin word to exist in two forms, of differing gender (even where biological gender is not a factor). The usual word for turnip (for example) is rapa, -ae (f), but rapum, -i (n) is also attested from antiquity. These are examples of what I call "collateral forms".
And a few words have different genders depending on the sense. Malus, -i (f), for example, means fruit tree, but malus, -i (m) means beam or mast.
There are still other areas of uncertainty about gender in Latin. To cite just one example: my signum computatorium is a feminine noun, but I am in fact biologically male. If I use an adjective to refer to me it should be masculine, but if I am referring to Iynx it should in form be feminine. Where does the boundary lie?
The Latin word for gender is genus. The ususal word for masculine (as a grammatical term) is masculinus, for feminine femininus, and for neuter neuter. (All these Latin words have other, non-grammatical senses as well, as does communis, used a little above).
I am trying, djmeyers, to pull this all together, and explain it more clearly, more tersely, and more thoroughly than it was ever explained to me. But I don't really know my audience. Am I making myself clear?