Subject and object in sentences

Nominative, genitive and partitive are the most important cases. They have special grammatical functions, that's why they are often called grammatical cases. The most important function is to distinguish the subject and the object.

Nominative

Subject is often in nominative case. Here koira (a dog) is in nominative, thus it's the object. These 2 sentences have the same meaning despite the word order:

Koira puri miestä.
Miestä puri koira.
The dog bit the man.

If mies (a man) is in nominative case, it will change the meaning.

Mies puri koiraa.
Koiraa puri mies.
The man bit the dog.

Now the man is the subject. Because the word order is quite free in Finnish, the case tells us who bit who.

Sometimes the object can be in nominative.

In commands: Tee varaus ajoissa! (Make a reservation in time!)
After a verb in passive: Laukusta varastettiin kännykkä. (A cell phone was stolen from a bag.)

The component after "olla" verb (to be) is called the predicative. Predicative should not be mixed with the object of the sentence. (see more about this sentence type). They describe the subject, and they are in nominative:

Talvi on kylmä.: Winter is cold.
Hanki on valkoinen.: The snowdrift is white.

Genitive (n-object)

Typically the subject is in nominative and object is either in partitive (see below) or genitive (some grammars also call it the "n-object").

Mies söi leivän.: A man ate the bread.

Partitive

Basic meaning of the partitive case is a part of something, or an undefinite amount. Quite often, if you can add "some" infront of it, you should use the partitive:

Pihalla oli lunta.: There was (some) snow outside.
Vesi oli kylmää.: The water was cold.
Saisinko vettä?: May I have (some) water?

Like the genitive, also the partitive can mark the object: the subject is in nominative case and the object eather in genitive or partitive. For example:

Mies söi leipää.: A man ate (some) bread.

Genitive or partitive?

The object of the sentence can be in genitive or partitive case. The typical difference is, if the process is finished or unfinished. If the process is unfinished, the object considers only a part of something, that's why it's in the partitive case:

Luin kirjan loppuun.: I read the book through.
Luin kirjaa illalla.: I was reading the book in the evening.

Another difference is between the positive and negative sentence: the object of the negative sentence is in partitive:

Ostan tuon kirjan.: I will buy that book
En osta tuota kirjaa.: I won't buy that book.

If the amount is indefinite, the object is in partitive:

Ostan pullon viiniä.: I'll buy a bottle of wine.
Ostan kaupasta viiniä ja leipää.: I'll buy (some) wine and bread from the shop.

The same idea applies also to the existential sentence ("there's something somewhere"): the subject of the sentence is in partitive, if it refers to some indefinite amount:

Lasissa on viiniä.: There's (some) wine in the glass.
Lompakossa on vähän rahaa.: There's little money in the wallet.

As opposed to:

Viini on lasissa.: The wine is in the glass. (All the wine is in the glass.)
Raha on lompakossa.: The money is in the wallet. (All the money is in the glass.)