Gender, more specifically known as grammatical gender or sometimes more aptly noun class, is a system in languages that divides nouns (and sometimes adjectives and other parts of speech) into various classes based on semantic or phonological criterion.
English does not have grammatical gender; instead it has a "natural gender" system, where the usage of gendered words (mostly pronouns, but also words like woman, man, girl, boy, waitress, waiter etc.) reflect physical or perceived sex or gender.
The systems most frequently referred to as gender have divisions such masculine, feminine, and neuter. This is true of Romance and (most) Germanic languages. It is important to note that grammatical gender does not necessarily have much to do with physical sex characteristics/ gender identity/ presentation etc; a clear example of this is in Old English, which had feminine, masculine, and neuter genders, and in which the words wif "wife," wifmann "woman," and laedig "lady" were feminine, masculine, and neuter respectively. These categories are little more than accidents of history, and the association of "masculinity" or "femininity" to these classes is nothing but a coincidence.
Another common system is animacy. Two levels of animacy are often observed (animate/inanimate), although more complex systems also occur (inanimate/ fluid/ animal/ human/ god, etc). There is often an animacy hierarchy wherein humans are at the top and objects such as rocks and dirt are at the bottom.
However, like grammatical gender, there is an element of arbitrariness to grammatical animacy as well; in Blackfoot, for instance, knives and drums are animate, whereas certain body parts are not.
In other languages, noun classes divide between shapes (long and skinny, short and round).