A lot of people make it to the level of being a stable player. They play confidently, at a steady tempo and never miss a note. This is admirable, but it is not artistry. A performance by a musician at this level is cold and robotic. To make her or his performances warm and relatable, a musician must learn to interpret a piece.
Interpretation is where the musician expresses how she or he hears, understands and feels about the piece. She or he does this by shaping the musical phases using the techniques of articulation, rubato and dynamics.
One form of articulation is how the notes connect to one another. Is it a smooth (legato) connection or are they detached (staccato)? How short are the notes when they are detached?
The other form is when a single note or small group of notes are louder than the surrounding notes.This is called a stress accent (or dynamic accent). How much louder are they played? Are they combined with legato or staccato playing?
Rubato involves varying the tempo to shape the phrase. One would usually start a phrase slightly slow, speed up so that the climax of the phrase is slightly fast and slow down to end the phrase slightly slow. It’s like a bird taking off, hitting top speed, then landing.
Dynamics are used to shape the phrase by starting out softly, growing louder so the the climax of the phrase is the loudest and then becoming softer. This is much like a rousing speech, starting off conversationally, reaching fever pitch and then finishing quietly.
Special attention is often given to dissonant notes and their resolution. The dissonance is usually played louder and held slightly longer than its notated value It is like the dissonance is being savored. The resolution is often the last note of the phrase and by playing it softer and giving it less time than the notated value, it is as if the phrase is dying.
The amount of variation in tempo and dynamics depends not only on who is playing the piece, but on the time period in which the piece was written. Baroque pieces would be played with the least amount of variation and romantic pieces would be played with the most.
These techniques breath life into a piece of music. The performance becomes more than the notes written on the score. It becomes a work of art. Each musician does this in a unique way. This is why one person may prefer the Elgar cello concerto as performed by Jacqueline Du Pré and another may prefer it as performed by Yo Yo Ma.
I recommend listening to the same piece played by a few different world class players and listen for how they interpret it. As you develop ears to hear it, you will have a deeper appreciation for the master musician.