Since the Baroque era, composers across Europe have mostly written performance instructions in Italian, regardless of their own nationality — a convention that's stuck ever since. These are the terms you'll run into most often, grouped by what they describe.

Tempo (speed)

Largo
Very slow and broad.
Adagio
Slow, at ease.
Andante
At a walking pace.
Moderato
A moderate, middle-of-the-road tempo.
Allegro
Fast and lively - one of the most common tempo markings in the entire repertoire.
Vivace
Lively and brisk, typically a shade faster than allegro.
Presto
Very fast.
Accelerando
Gradually speeding up.
Ritardando (rit.)
Gradually slowing down.
A tempo
Return to the previous, original tempo after a deviation.

Dynamics (volume)

Pianissimo (pp)
Very soft.
Piano (p)
Soft.
Mezzo piano (mp)
Moderately soft.
Mezzo forte (mf)
Moderately loud.
Forte (f)
Loud.
Fortissimo (ff)
Very loud.
Crescendo (cresc.)
Gradually getting louder.
Diminuendo / Decrescendo (dim. / decresc.)
Gradually getting softer.

Our dynamics and tempo markings guide goes deeper on how these two categories actually work in practice.

Articulation & expression

Legato
Smoothly connected, with no gap between notes.
Staccato
Short and detached, each note clearly separated.
Fermata
Hold the note or rest longer than its written value, at the performer's discretion.
Da capo (D.C.)
Return to the beginning of the piece.
Dal segno (D.S.)
Return to an earlier marked "sign" in the score, rather than the very beginning.
Cadenza
A soloist plays alone, often improvisatory in feel - most familiar from the concerto.
Con brio
With vigor and spirit.
Dolce
Sweetly, gently.
Espressivo
Expressively, with emotional emphasis.