This is another in the series of Postgres' unique capabilities, which you might not find in other RDBMS products (sometimes not even in commercial products!).
I have been a fan of partial indexes since I came across them about 3 years ago, and to show the utility of this unique feature here's a post from Command Prompt's JD explaining a unique use-case and how Partial Indexes can be used to solve the problem:
The post above does not go into details of how partial indexes solve the problem (JD even mistakenly calls it expression-indexes), and neither will I try to explain it since TFM at postgres.org explains it quite well:
All hail Postgres.