ORDER BY gives ERROR: function array_position(text[], character varying) does not exist
Sample data

You have a table like,

CREATE TABLE research(colors)
  AS VALUES ('Blue'), ('Orange'), ('Yellow');


ENUM type

You have an enumerated list of colors. So the easy thing here would be to use an ENUM type

CREATE TYPE colors AS ENUM ('Red','Orange','Yellow','Green','Blue');


Then

ALTER TABLE research
  ALTER COLUMN colors      -- myColorsColumn
  SET DATA TYPE colors
  USING (colors::colors);  -- myColorsColumn::NewType


Now it's faster, more efficient, and cleaner. MORE WIN. MORE JOY. Etc. ENUM are stored as 4-byte internally.

ORDER BY colors


Is all you'll need.

Array Sort

But, you've got a God-given right to treat this like any other database that doesn't support ENUM types,

ORDER BY array_position(ARRAY['Red','Orange','Yellow','Green','Blue'], color);


I think this is perfectly fine, but I believe there is a limitation here in the implementation,

ARRAY['foo', 'bar', 'baz']


Is essentially

ARRAY['foo', 'bar', 'baz']::text


This means you need to either (speed doesn't matter as both options perform the same)


Make the color column of the native text. This can be done in the call, or you can actually modify the table and should. In PostgreSQL, nothing should be varchar without a limit (since that's just a parallel type for text), and very few things should be varchar with a limit (since there is no advantage)

color::text            -- PostgreSQL sexy sexy cast
CAST(color AS TEXT))   -- ANSI SQL standardized vanilla and boring

Or, you can construct the array itself as type varchar[]

array_position(ARRAY['Red','Orange','Yellow','Green','Blue']::varchar[], color);



Further notes


In PostgreSQL, we wouldn't use varchar for colors. Even if you insist on not using an ENUM here (though I would), that should be text
The array_position is a shorthand, I expect it to be substantially slower than a similar operation though

CASE
  WHEN color='Red' THEN 1::smallint
  WHEN color='Orange' THEN 2::smallint
  WHEN color='Yellow' THEN 3::smallint
  ... etc
END;