Taken from the Music Publisher's Association website - which also has short videos and a list of registered publishers.

The business of music publishing is concerned with developing, protecting and valuing music. The business is diverse and demands a variety of skills. Music publishers play a vital role in the development of new music and in taking care of the business side, allowing composers and songwriters to concentrate on their creative work.
The Music Publisher's Association definition of a good publisher

"A good publisher seeks out great music and great composers and songwriters; supports composers and songwriters in the creative process; promotes their catalogues across a variety of platforms; manages the business exploitation of the catalogues (including the registration of works and the collection and onward payment of all due royalties); and generally seeks to protect and enhance the value of their works with passion and professional commitment."

What does a music publisher do?

The business is diverse and demands a variety of skills. The role of a music publisher involves:

Finding new and talented songwriters and composers and encouraging and supporting them as they develop their skills, whether through helping with their living expenses, providing them with the facilities they need to produce music or offering advice and guidance in writing for particular markets.

Securing commissions for new works and helping to coordinate work flow.

Registering the works of songwriters and composers with all appropriate collecting societies and agencies, eg PRS for Music in the UK.

Producing demo recordings and, in the case of contemporary classical music, performance materials (score and parts etc).

Producing and licensing the production of printed music.

Producing promotional materials, including sampler CDs, study scores, brochures etc.

Promoting composers and songwriters and their music to performers, broadcasters, record companies and others who use music on a commercial basis, both nationally and internationally.

Licensing the use of music, whether directly in the case of individual and special usages (eg synchronisation deals) or via the collecting society network.

Responding to new licensing opportunities that flow from technological developments.

Monitoring and tracking the use of the music they own and ensuring that proper payment is made for all licensed uses.

Making royalty payments to songwriters and composers in respect of the usage of their music.

Taking appropriate action against anyone using music without the necessary licence.