Postgresql SELECT if string contains

A proper way to search for a substring is to use position function instead of like expression, which requires escaping %, _ and an escape character (\ by default):

SELECT id FROM TAG_TABLE WHERE position(tag_name in 'aaaaaaaaaaa')>0;

You should use tag_name outside of quotes; then it's interpreted as a field of the record. Concatenate using '||' with the literal percent signs:

SELECT id FROM TAG_TABLE WHERE 'aaaaaaaa' LIKE '%' || tag_name || '%';