When we say “It is raining” in English, what is this “it”? Well, it is just a formal subject, used because the English grammar says so. In Finnish, we say simply Sataa, which is a verb form, corresponding to “rains”. No subject is expressed or implied, but the verb is still in a personal form, 3rd person singular. Similarly, in English we say “It is sad that…”, but in Finnish On ikävää, että…, using the verb form on (is) without a subject. We say “It is warm here”, but in Finnish it’s just Täällä on lämmintä.
Thus, when a verb expresses an event or state of nature, which is normally not considered to have an agent of any kind, Finnish simply uses the verb without a subject.
Somewhat arguably, the verb sataa may have a subject. We can say Sataa lunta (It is snowing; literally: Rains snow). Here lunta, the partitive of lumi (snow), can be interpreted as a subject or as an object. In the latter case, the sentence lacks a subject.
Finnish also uses clauses without a subject to express states of mind or body. For example, päätäni särkee (I have headache) can be interpreted so that päätäni (my head; partitive form) is the object, and the predicate särkee (breaks, hurts) has no subject. Many causative verbs are used this way, e.g. Minua väsyttää (I am tired).
In English, presence clauses like “There is an apple on the table” have the formal construct “there is”, which is somewhat similar to a formal subject. In Finnish, such constructs are not used, and presence clauses have a different word order: the thing that is present is mentioned last, as it the new thing: Pöydällä on omena, literally “on table is apple”. This has the same structure as an expression for having something; normally the context and meaning resolve the ambiguity.
There is yet another type of sentences without a subject: those that imply a generic person, much like English uses the pronoun “you” in a generic meaning. These are discussed in section Generic statements.
Verbs that express causation of sensations, feelings, etc., are very often used without a subject. We can say Minua itkettää (I feel like crying) or simply Itkettää, implying the object. The verb itkettää is a causative derivation of itkeä (to cry) and thus means “to make (someone) cry”. It can be used with a normal subject, expressing what makes someone cry, e.g. Uutinen itketti minua (The news made me cry). More often, however, the subject is completely omitted. This does not just avoid mentioning the thing that causes crying; it also focuses on the effect, someone feeling like crying, rather than causation.
In literary style, a verb form often implies a subject, e.g. näen (I see) implies the subject minä. Such sentence do not really lack a subject, they just have it as implicit.
Somewhat arguably, this applies to 4th person forms like nähdään, too, though the implied subject is vague.