The basic meaning of the Partitive case is "partialness", "without result" or "without specifying identity".

In the Finnish language, it's used to express unknown identities and irresultative actions. For example, it is found in the following circumstances, with the characteristic ending of "a" or "ta":

• After numbers: "kolme taloa" -> "three houses"
• For incomplete actions and ongoing processes: "luen kirjaa" -> "I'm reading a book"
• After certain verbs, particularly those indicating emotions (as they are irresultative): "rakastan tätä taloa" -> "I love this house"
• For tentative enquiries: "saanko lainata kirjaa?" -> "can I borrow the book?"
• In places where English would use "some" or "any": "onko teillä kirjoja?" -> "do you have any books?"
• For negative statements: "talossa ei ole kirjaa" -> "there is not a book in the house"

A good example of the irresultative meaning of the partitive is this: ammuin karhun (accusative) means "I shot the bear (dead)", whereas ammuin karhua (partitive) means "I shot (at) the bear" without specifying if it died. Notice that Finnish has no native future tense, so that the partitive provides an important reference to the present (luen kirjaa) as opposed to the future (luen kirjan). The latter means "I will read the book", as a result ("the book has been read") indicates action in the future.

The case with an unspecified identity is onko teillä kirjoja, which uses the partitive, because it refers to unspecified books, as contrasted to accusative onko teillä (ne) kirjat?, which means "do you have (those) books?"

It is important to use the partitive with the numerals, because the incorrect form without the partitive, *kymmenen kirjat, would translate to "ten (those) books", which is nonsensical.

Reference: http://virtual.finland.fi/netcomm/news/showarticle.asp?intNWSAID=25838