Short answers, pulled from the story.
Grammatical tense is the category that expresses time reference, placing a state or action at a time relative to the moment of speaking. It is usually shown through specific verb forms, particularly conjugation patterns. In modern theory it grammaticalizes time reference, while traditional European grammar often stretched the term to cover aspect and mood as well.
The number varies widely. Some languages have only two tenses, such as past and nonpast or future and nonfuture, while others have all three of past, present, and future. Kalaw Lagaw Ya of Australia has six, separating the remote past, recent past, today past, present, today or near future, and remote future, and some languages have no tense at all.
Tense expresses time reference, aspect expresses how a state or action relates to time such as complete versus ongoing, and mood expresses modality including uncertainty, evidentiality, and obligation. When the three do not separate cleanly in a verb form, a language can be described as having a single tense-aspect-mood, or TAM, system.
Tenseless languages include Burmese, Dyirbal, most varieties of Chinese, Malay including Indonesian, Thai, Yucatec Maya, and Vietnamese, with Greenlandic and Guaraní tenseless in some analyses. They still refer to time using lexical items like time phrases, aspect markers, and context, as Mandarin does with the markers le and guò.
The historical present is the use of the present tense to refer to past events, showing that a tense form does not always carry its basic time meaning. Fake tense is the broader crosslinguistic phenomenon of using a tense form, often a past form, to mark counterfactuality in conditionals and wishes.
Early grammarians, often monks, had no reference point other than the Classical languages to describe their own, so Latin terminology spread to modern languages, sometimes with changed meaning. Latin was traditionally described as having six tenses, though a newer grammar argues it really has three, present, past, and future, with the rest being aspect.