Rhythm encompasses the idea of time as it relates to music. The specific beat, time signature and tempo of a song all fall under the element of rhythm. More complex aspects of rhythm include syncopation and tempo changes.
Melody in music is a sequence of pitches that often acts as a theme in a piece of music. Melodies are built around musical scales, and often repeat throughout a piece of music. Melodies are generally described as conjunct, which means smooth sounding and easy to play, or disjunct, which means that the melody has large pitch jumps and is harder to play.
Harmony is the expanding of the melody into chords and accompaniments. If you look at a piece of music, you can see that melody is the horizontal movement of notes while harmony is the vertical stacking of notes. Harmonies and the chord progressions that derive from them are based on the same scales as melodies.
Dynamics is the relative loudness or quietness of a section of music. A great deal of musical expression can be achieved simply via use of dynamics. Italian terms such as "piano" (meaning quiet) and "forte" (meaning loud) are used to describe dynamics.
Timbre is the specific sound that a certain instrument produces. It is because of musical timbre that we can tell the difference between an "A" note played on a trumpet, saxophone or a piano. Timbre is sometimes called the "tone color" of a musical instrument.
Texture refers to the way that the various individual musical lines in a song work with each other. Textures range from monophonic -- one melody line with no harmony -- to homophonic -- one melody accompanied by one harmonizing accompaniment -- to polyphonic -- multiple melodies being sounded at once.
The form of a piece of music is basically the way that all of the parts are put together. Letters usually designate certain parts of a song, so it's possible to spell the form using letters. For example, A B A (known as ternary form) is a common form of music in which one section is played, followed by a totally different section, followed by a repeat of the first section.