Granting access to all tables for a user
First, you have to be able to connect to the database in order to run queries.  This can be achieved by

REVOKE CONNECT ON DATABASE your_database FROM PUBLIC;

GRANT CONNECT
ON DATABASE database_name 
TO user_name;


The REVOKE is necessary because


  The key word PUBLIC indicates that the privileges are to be granted to
  all roles, including those that might be created later. PUBLIC can be
  thought of as an implicitly defined group that always includes all
  roles. Any particular role will have the sum of privileges granted
  directly to it, privileges granted to any role it is presently a
  member of, and privileges granted to PUBLIC.


If you really want to restrict your user to DML statements, then you have a little more to do:

REVOKE ALL
ON ALL TABLES IN SCHEMA public 
FROM PUBLIC;

GRANT SELECT, INSERT, UPDATE, DELETE
ON ALL TABLES IN SCHEMA public 
TO user_name;


These assume that you will have only one schema (which is named 'public' by default).

As Jack Douglas pointed out, the above only gives the privileges for the already existing tables.  To achieve the same for future tables, you have to define default privileges:

ALTER DEFAULT PRIVILEGES 
    FOR ROLE some_role   -- Alternatively "FOR USER"
    IN SCHEMA public
    GRANT SELECT, INSERT, UPDATE, DELETE ON TABLES TO user_name;


Here, some_role is a role that creates the tables, while user_name is the one who gets the privileges.  Defining this, you have to be logged in as some_role or a member of it.

And, finally, you have to do the same for the sequences (thanks to PlaidFan for pointing it out) - here it is the USAGE privilege that you need.