The word 'ode' comes originally from the Greek word aoide which means 'song'. Certainly the roots of the form are based in song, with the original Greek odes designed to match music and dance. Modern odes keep only the lyrical quality of this by maintaining regular meter and rhyme.
The main features of odes are that they are generally about important or dignified subjects that are addressed to the subject itself - Ode to the West Wind, Ode to a Nightingale, Ode on a Grecian Urn.
The original odes, known as Pindaric odes, named after the Greek poet Pindar, are composed in three sections, the strophe, antistrophe and epode. The strophe has a complex structure, the antistrophe is composed in a similar structure and the epode was structured differently to end the poem. The reason for this structure was to enable dance and music to accompany the reading of the poem. Pindar's odes were often celebrations of events, occasions and victory. The oldest Pindaric ode known in modern times is the '10th Pythian ode', which is about the victory of a man named Thessalian Hippocleas in a running race.
The Roman poet Horace is more influential to our modern understanding of odes. Horace's odes are less dramatic than Pindaric odes and are written more to be read than performed like the Pindaric odes. It is from Horace that subject matter of a serious nature became the basis of the form of odes. Horace's ode 4.7, for example, is an ode to spring through which Horace describes the joy of life and also the impermanence of it. He implores the reader to enjoy spring (life) before the seasons change to winter (death). This example demonstrates Horace's influence on the form. Many examples, particularly from the Romantic poets, deal with similar subject matter. Shelley's Ode to the West Wind, for example, deals with the transience of the seasons and the rebirth and new life that spring represents.
The structure of modern odes is decided upon by the composer. What is constant is the dignified and reverent subject matter, the point of view that addresses the subject matter, the serious tone and the high sense of emotion that the composer conveys to the reader.
Even then, some modern poets subvert the ode form and compose 'odes' about subject matter that is not serious. The poet uses a highly ironic tone to applaud the characteristics of a relatively insignificant subject.
Meter and Rhyme
While specific conventions of meter and rhyme are not a characteristic of odes, composers of odes nonetheless need to choose regular rhyme and meter. That is to say that odes do have consistent and regular meter and rhyme within each particular ode, but the exact rhyme and meter is at the composer's discretion.
What you will notice is that typical odes have a stanza length, a meter and a rhyme scheme that is adhered to rigidly. It is this characteristic that gives the form its song-like quality. Think about the songs that you listen to. The verses are usually the same length throughout the song.
Consider the example To Autumn by John Keats - online lesson to see the type of meter and rhyme that composers of odes often employ.