It looks like you’ve pulled up the Tempo Palette from a music notation software (like MuseScore or Sibelius). These marks are the "speed limit" signs for a piece of music.
They generally fall into three categories: Specific Beats, Descriptive Italian Terms, and Changes in Speed.
The symbols like ♩ = 80 are the most precise. They tell you exactly how many beats to play per minute (BPM).
♩ = 80: Play 80 quarter notes per minute.
Examples in your image: You see different note values (eighth notes, half notes, dotted notes). This tells the performer which specific pulse to count at 80 BPM.
Classical music uses Italian because it was the "universal language" of music for centuries. They are listed in your image roughly from slowest to fastest:
Term
Meaning
Feel
Grave / Largo
Very Slow
Solemn, heavy, or broad.
Adagio
Slow
Easeful and leisurely.
Andante
Walking Pace
A moderate, flowing speed.
Moderato
Moderate
Not too fast, not too slow.
Allegro
Fast
Bright, cheerful, and quick.
Presto / Prestissimo
Very Fast
A total sprint; as fast as possible.
These tell you to change your speed while you are already playing:
rit. (ritardando): Gradually slow down.
accel. (accelerando): Gradually speed up.
rall. (rallentando): Similar to ritardando; a "dying away" or widening of the tempo.
a tempo: Return to the original speed (usually after a rit. or accel.).
tempo primo: Go back to the very first tempo of the piece.
Swing / Straight: Common in Jazz or Pop. "Swing" gives the music a "long-short" bouncy rhythm, while "Straight" means play the notes exactly as written.
♩ = ♩ (Metric Modulation): These symbols are used when the time signature changes. They tell the pianist to keep the pulse the same even if the "look" of the notes changes.
Pro Tip: If you see -etto or -ino added to a word (like Allegretto), it usually means "a little bit less." So, Allegretto is slightly slower and less intense than a full Allegro.
Would you like me to find a specific song example for any of these so you can hear what they sound like?