How to join two tables using a comma-separated-list in the join field

select
    m.id,
    group_concat(c.name)
from
    movies m
    join categories c on find_in_set(c.id, m.categories)
group by
    m.id

The output should be something like this:

Table movies {
  -id-       -categories-
  1          Comedy,Drama
  2          Action,Drama
  4          Other,Dance
}

Using comma separated lists in a database field is an anti-pattern and should be avoided at all costs.
Because it is a PITA to extract those comma separated values out agian in SQL.

Instead you should add a separate link table to represent the relationship between categories and movies, like so:

Table categories
  id integer auto_increment primary key
  name varchar(255)

Table movies
  id integer auto_increment primary key
  name varchar(255)

Table movie_cat
  movie_id integer foreign key references movies.id
  cat_id integer foreign key references categories.id
  primary key (movie_id, cat_id)

Now you can do

SELECT m.name as movie_title, GROUP_CONCAT(c.name) AS categories FROM movies m
INNER JOIN movie_cat mc ON (mc.movie_id = m.id)
INNER JOIN categories c ON (c.id = mc.cat_id)
GROUP BY m.id

Back to your question
Alternativly using your data you can do

SELECT m.name as movie_title
  , CONCAT(c1.name, if(c2.name IS NULL,'',', '), ifnull(c2.name,'')) as categories 
FROM movies m
LEFT JOIN categories c2 ON 
 (replace(substring(substring_index(m.categories, ',', 2),
  length(substring_index(m.categories, ',', 2 - 1)) + 1), ',', '') = c2.id)
INNER JOIN categories c1 ON 
 (replace(substring(substring_index(m.categories, ',', 1), 
  length(substring_index(m.categories, ',', 1 - 1)) + 1), ',', '') = c1.id)

Note that the last query only works if there are 2 or fewer categories per movie.