In the blues, and in many other styles of music, most notably jazz, a swing rhythm is a rhythmic concept that elongates the first note and shortens the second. This subdivision effectively “swings” the notes. When this “swing” rhythm is repeated multiple times, one directly after the other, the effect is a “shuffle”. A shuffle is simply a “groove” of repeatedly “swung” notes.
A straight rhythm is often notated plainly, with eighth notes appearing one after the other with not modulation. You will usually see no accompanying directions besides that the tune is to be played “straight.”
Shuffle rhythms are notated in western music as triplets. The first note in the triplet is twice the length of the second note, leaving the impression of an elongated first note and shortened second note. Below is a literal notation of 4 swung pairs of eighth notes. The repetition of swung notes is what makes a shuffle.
Swing rhythm isn’t as pronounced as shuffle rhythm. In a shuffle, the backing instrument will play a continuous stream of swung eight notes. In swing, a bass player may play a walking quarter note bassline while the drummer taps out a simple swing rhythm.
Swing and shuffle in common sheet music
It helps to think of swing of shuffle rhythms as “feels” or “grooves” and not direct notation styles. Often you’ll find that a lead sheet will note notate swing rhythms at all, but will show a small text note at the top of the page that reads “Medium Swing” or “Shuffle.”
The reason composers or transposers don’t literally notate swing rhythms is because they tend to become very messy and difficult to read. Most proficient musicians can simply take an otherwise straight piece of music and impose different “grooves” into the music.