If you are looking up names or looking for simple strings, you are usually querying the entire content of a field. In full-text search, this is different. The purpose of the full-text search is to look for words or groups of words that can be found in a text. Therefore, full-text search is more of a contains operation, as you are basically never looking for an exact string.
In PostgreSQL, full-text search can be done using GIN indexes. The idea is to dissect a text, extract valuable lexemes (= "preprocessed tokens of words") string, and index those elements rather than the underlying text. To make your search even more successful, those words are preprocessed.
Here is an example:
test=# SELECT to_tsvector('english',
'A car, I want a car. I would not even mind
having many cars');
to_tsvector
---------------------------------------------------------...