Normally, the subject is in the nominative, e.g. Lapset leikkivät puistossa (The children are playing in the park). This is the most common use of the nominative, but the nominative is used in some other contexts as well.
When the subject is in the nominative and it is placed at the start of the sentence, it is usually definite, i.e. we would use the article “the” in English.
In a presence clause in a broad sense, the subject is in the partitive, except when all the following conditions are met:
Examples:
“Uncountable” things typically express a substance, e.g. Kupissani on teetä, ei kahvia (There is tea, not coffee, in my cup) and Taskussani on paljon rahaa (There is a lot of money in my pocket).
In questions asking about presence, the partitive is used even in singular, e.g. Onko tilalla hevosta? (Is there any horse in the farm?). The question Onko tilalla hevonen? is possible, too, but has a special meaning: it more or less implies that there is a horse in the farm, or at least strong evidence on it, and it expresses surprise about the situation, somewhat like “Oh, is there a horse in the farm, really?”
Somewhat differently, the question Tuleeko hänestä opettaja is neutral: “Will he become a teacher?”, whereas Tuleeko hänestä opettajaa expresses strong doubts about the matter, corresponding to “I wonder if the will ever become a (real) teacher”. This differs from the preceding type of questions in being about becoming something, rather than being something.
In some grammars (including Iso suomen kielioppi), expressions like minun täytyy nyt mennä (I must go now) are analyzed as having the subject, minun, in the genitive. This may sound natural, but it is more logical to treat them as subjectless sentences and an expression like minun as an adverbial. Thus, in the example, minun can be interpreted as meaning “to me”. Note that the adverbial in genitive, like minun in the example, may be absent, e.g. Nyt täytyy mennä; this leaves it open (normally to be determined by context) to whom the statement applies.
There are similar issues with e.g. the verb pitää (must, need) as well as expressions with on pakko (it is obligatory) and on lupa (it is permitted) and similar expressions, like on hyvä (it is good/recommended), on paras (it is best), and on syytä (there is a reason to). Such things are normally described in good dictionaries in an entry for the verb or (when the expression involves the verb form on) the adjective or the noun. In expressions like on hyvä, the verb is always in 3rd person singular form, e.g. meidän on pakko (we have to), heidän oli pakko (they had to). The reason is that the grammatical subject is the infinitive of a verb, e.g. in miesten oli pakko lähteä (the men had to leave) the subject is lähteä.