In English you can conjugate like so:
I eat
You eat
He/she/it eats
We eat
You all eat
They eat
But you can also conjugate with a variety of “indefinite” pronouns:
One eats
Everyone eats
No one eats
Few eat
Some eat
Many eat
Etc.
I’m wondering what this would look like in Latin and how you would categorise it in terms of person and number? E.g., is “one eats” first, second or third person? Is it singular or plural?
The rules in Latin are somewhat the same as in English: use a separate word for the subject, then conjugate the verb to agree with it.
In English, you say "one eats" but "many eat" because the former is third person singular and the latter is third person plural; in Latin, the same applies: aliquis edit, multī edunt.
Direct translations of the words you ask for are aliquis "someone", omnis "every person", nemō "no one", paucī "a few", aliquī "some people", and multī "many". But you can really use any adjective or noun like this: ūnus "exactly one person", duō "two people", malī "the evil people", and so on.
In addition to Draconis's suggestions, you can use the passive voice to express an indefinite 'agent':
Romam itur. "[indefinite subject / 'it'] is being gone to Rome" => "One goes to Rome", in context probably "We/I/they/etc. go to Rome".
In horto esum est. "One ate in the garden".
Noctis pugnandum est. "One ought to fight at night".
In addition to the other answers, on specific conditions, one can also express an indefinite person using the present subjunctive(*) of the 2nd person singular. In a manner not not unsimilar to "Generic you" in English and other languages.
According to A&G (§518) in general conditions :
The subjunctive is often used in the 2nd person singular, to denote the act of an indefinite subject (you = any one). Here the present indicative of a general truth may stand in the apodosis.
Virtūtem necessāriō glōria, etiamsī tū id nōn agās, cōnsequitur. (Tusc. 1.91) Glory necessarily follows virtue, even if that is not one's aim.
For a detailed reading, one can consult: Hale, William Gardner. "An Unrecognized Construction of the Latin Subjunctive: The Second Person Singular in General Statements of Fact." (1906).
(*) Several sources (like of William Gardner) claim that the indicative can be also used in general statements. e.g:
Bis peccas cum peccanti obsequium commodas. ( Publil. 52.). You sin doubly when you humour a sinner
