Tempo is a very important and often controversial element in classical music. Tempo is simply the Italian word for time, and refers to the basic rate of speed or slowness in a piece of music. The plural is Tempi.
For centurires, composers have indicated the basic speeds they wanted in any given piece of music, and the most commonly and universally used terms have been Italian, although German , French, English and American composers have sometimes used their own languages for these indications, such as Debussy and Ravel, who used French equivalents of Italian terminology. Along with the indications for rates of speed, composers often added further Italian , French or German indications for the expressive character they want the music to convey.
If you compare different recordings and live performances, you will notice that different musicians often use markedly different tempi ; slow and fast are relative terms, and there are many degrees of slowness and speed. If you compare the timings of a lengthy symphony on different recordings, one may be up to ten or more minutes longer than another.
Tempo has caused a great deal of controversy of the years ; different scholars and experts argue endlessly over what the right and wrong tempi are for the music of composer X or Y, or for the music of one period or another. So do fans. Critics will often lambaste a particular musician, a conductor or pianist etc, for tempi that seem to him or her too fast or too slow. It's all highly subjective. Composers have also been upset by musicians who use tempi they consider wrong.
In the early 19th century, Austrian inventor Johann Nepomuk Maelzel invented the metronome, a device by which composers could indicate the precise temi they wanted, by indicating x beats per minute. 60, 80, 100 etc. This was not meant to cause regidity of tempo, but basic rates of speed ; performers were expected to use Rubato, or spontaneous modifications of temp for expressive reasons. Later, metronomes were used to keep practicing students from dragging or rushing.
The first great composer to use metronome markings was Beethoven, and in our time many musicologists and critics have used these markings as an excuse to blast musicians for not following them. But the problem is that composers change their minds, and have been known to disregard their own metronome markings when conducting or playing their music at later dates. Igor Stravinsky ( 1882 - 1971 ), was a stickler for performers observing his markings.
Italian tempo indications range from Presto - very fast, or Prestissimo, REALLY fast, Vivace, or lively, Allegro, or moderately fast, Andante, lierally going, or neither fast nor slow, and slow tempi such as Adagio, Lento and Grave (grah-veh ). Moderato, or moderate. Larghissimo, or very slow etc. There are other relative terms such as allegretto, Andantino, Allegro Vivace, etc.
There are other terms for altering the tempo, making it gradually faster or slower, Accelerando, or Ritardando, Ritenuto or suddenly slower, Piu Moto, or just go faster etc. Allargando does not mean to play in a reptilian manner but to gradually broaden the tempo.
There are other indications such as Allegro non Troppo, or not too fast, Assai, or very, Molto, or very much, Poco, or a little, Subito, or suddenly, Meno Mosso, or just slower, Con, or with, as in Con Fuoco (with fire, an expressive indication ), Senza, or without, Quasi, or as if, Poco a Poco, or little by little, A Temp, or return to the previous tempo, etc.
Expressive markings are often added to terms such as Andante, Allegro etc. Tranquillo, or tranquilly, Agitato or agitated, Dolce, or sweetly, (Dol- cheh ), Giocoso, or merrily, Misterioso, Con Brio, Cantabile , in a singing manner, etc. Beethoven, Wagner, Richard Strauss, Paul Hindemith and other German and Austrian composers often used their own language, using terms such as Langsam, or slow, Lebhaft, or lively, Rasch or fast etc.
One problem with recordings is that if you get familiar with a particular recording of a Beethoven, Brahms or Tchaikovsky symphony etc, you will get accustomed to the tempi of that particular performance, and when you hear other recordings, the tempi may seem wrong to you. But I've found that if you continue to listen to other performances, you can get accustomed.
Fan X may argue with Fan Y that conductor X's tempi are wrong and conductor Y's are right, but who knows what the composer would have wanted ? We have recordings of great composers such as Stravinsky, Richard Strauss, Paul Hindemith, Aaron Copland, Benjamin Britten and others conducting their own music, but even the composers might agree that their way is not the only one. In cases where a composer made more than one recording of the same work, the tempi may be different !