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a Conducting in the musical sense means: beating time to help a group of musicians to play well together.
If a large orchestra is playing music, it is important that they all play exactly together. They need to know exactly when to start, what speed to go, how loud or quietly to play and what the mood of the music should be. If two, three or four people play music together they can talk about this amongst themselves and one person can nod with his/her head or with a violin bow or flute to help the group to start and finish together. With an orchestra there are so many people that they need a conductor.
In the 17th century orchestras were usually small so they did not need a conductor. Often they were directed by the keyboard player or lead violin. But as orchestras grew in size and complexity the convention emerged of having non-playing musician stand facing the orchestra as director/conductor. The French composer Lully (1632-1687) used to beat time by banging a big stick (like a walking stick) on the floor to the time of the music. One day he banged his stick very hard and it went through his foot and he died of gangrene.
Conducting as we know it had become normal by the 19th century. The composer Mendelssohn (1809-1847) was a very good conductor. Some conductors in Victorian times were very conceited and behaved like showmen. The conductor Louis Antoine Jullien (1812-1860) was a French conductor who often came to England. He wore white kid gloves which were presented to him on a silver tray at the start of the concert. He dressed in expensive clothes and his long black hair waved all over the place as he conducted. His success was immense, in France at first, in the UK afterwards and then even in the US where he worked with the showman P.T. Barnum. His concerts were a mix of dance and "classical" music, always with the best musicians. his life is so peculiar that a biography (in French) has been published (see http://louisjullien.site.voila.fr). The conductor Sir Henry Wood (1869-1944), who was famous for conducting The Proms was a well-liked man who was respected and loved by orchestras and audiences.
Technique of conducting
Conductors usually beat time with their right hand. This leaves their left hand free to show the various instruments when they have entries (when they start playing) and to give interpretative gestures such as indicating when to play louder or softer. Most conductors have a stick called a “baton”. It makes it easier for people at the back of large orchestras or choirs to see the beat. Other conductors prefer not to use a baton. A conductor stands on a small platform called a “rostrum”.
To be a good conductor is not easy. It is not just a question of giving a steady beat. A good conductor will know the music extremely well, understand the composer's intentions, understand all technical details and know how to be able to work with the orchestra to create great music. Having good communications skills is an asset but some conductors speak very little during their rehearsals. They make everything clear through the way they conduct.
Some of the most famous conductors of the past were: Gustav Mahler, Hans Richter, Arthur Nikisch, Arturo Toscanini, Bruno Walter, Wilhelm Furtwängler, Herbert von Karajan, Leopold Stokowski, Georg Solti, John Barbirolli, Otto Klemperer, George Szell and Leonard Bernstein.
The main conductor who is in charge of an orchestra is often given the title "musical director". This will usually mean that s/he has a lot of power in the organization of the orchestra. Orchestras may give honorary titles to their conductor such as "conductor laureate". A "guest conductor" is one who conducts an orchestra regularly but is not the main conductor. An "assistant conductor" will often be a young conductor who helps the main conductor and gets the chance to conduct some of the concerts.