Trying to count multiple values and total
Try this for a count of competitions in each genre:

SELECT 
  COUNT(competition.competition_id) as competition_count, 
  genre.genre_id,
  genre.name
FROM 
  genre
  INNER JOIN competition on ( genre.genre_id = competition.genre_id )
  INNER JOIN band on ( band.band_id = competition_band.band_id )
GROUP BY genre.genre_id, genre.name
ORDER BY genre.name;


A count of bands who have entered competitions sounds like a different query to me:

SELECT 
  COUNT(band.band_id) as bands_in_competitions_count, --double counts bands in multiple competitions
  COUNT(distinct band.band_id) as bands_in_competitions_in_this_genre,
  genre.genre_id,
  genre.name
FROM 
  genre
  INNER JOIN competition on ( genre.genre_id = competition.genre_id )
  INNER JOIN competition_band on ( competition.competition_id = competition_band.competition_id )
  INNER JOIN band on ( band.band_id = competition_band.band_id )
GROUP BY genre.genre_id, genre.name
ORDER BY genre.name;