Question

I want to know what PostgreSQL functions are.
When do I have to write them?
How can I write them?
And how can I call them?

Answer 1

Definition, from wikipedia:

A stored procedure is a subroutine available to applications that
access a relational database system.

Advantages of stored procedures in general, from wikipedia:

Overhead: Because stored procedure statements are stored directly in
the database, they may remove all or part of the compilation overhead
that is typically required in situations where software applications
send inline (dynamic) SQL queries to a database. (…)

Avoidance of network traffic: A major advantage with stored procedures
is that they can run directly within the database engine. In a
production system, this typically means that the procedures run
entirely on a specialized database server, which has direct access to
the data being accessed. The benefit here is that network
communication costs can be avoided completely. This becomes
particularly important for complex series of SQL statements.

Encapsulation of business logic: Stored procedures allow programmers
to embed business logic as an API in the database, which can simplify
data management and reduce the need to encode the logic elsewhere in
client programs. (…)

Delegation of access-rights: In many systems, stored procedures can be
granted access rights to the database that users who execute those
procedures do not directly have.

Some protection from SQL injection attacks: Stored procedures can be
used to protect against injection attacks. Stored procedure parameters
will be treated as data even if an attacker inserts SQL commands. (…)

In PostgresSQL stored procedures are called user defined functions.
Definition example:

CREATE FUNCTION somefunc(quantity integer) RETURNS integer AS $$
DECLARE
    myvariable integer := 2;
BEGIN   
    RETURN quantity * myvariable;
END;
$$ LANGUAGE plpgsql;

(You can use other languages to define stored functions in PostgreSQL)

Calling example:

SELECT somefunc(100);

More info: http://www.postgresql.org/docs/9.1/static/server-programming.html