A partial index can e.g. be used to enforce a unique constraint on only a subset of the data.

The rule "a project name must be unique among all active projects" can be enforced with such an index:

create table projects
(
   project_id integer not null primary key,
   name       varchar(100) not null,
   is_active  boolean;
);

create unique index idx_unique_active_name 
   on projects (name)
   where is_active
	

With the above definition, the following rows can be inserted

insert into projects (project_id, name, is_active) values (1,'WebShop', true);
insert into projects (project_id, name, is_active) values (2,'CRM System', true);
insert into projects (project_id, name, is_active) values (3,'WebShop', false);

But the following insert will fail:

insert into projects (project_id, name, is_active) values (1,'CRM System', true);

As a side effect the size of the index is reduced if only few rows match the WHERE clause of the index.

More information about partial indexes can be found here: http://use-the-index-luke.com/sql/where-clause/partial-and-filtered-indexes