Tchaikovsky wrote three program music works based on William Shakespeare’s plays, among which Romeo and Juliet overture was undoubtedly the most famous; the other two are far less known, one of them is this symphonic poem (fantasia) “The Tempest”, and Hamlet overture, Op. 67, which is probably the least known. I only heard this piece recently, and I thought this was such a beautiful piece, and much underrated.
It’s probably worth putting down the program text printed on the original score here as a guide of listening:
The Sea. The magician Prospero commands his spirit Ariel to create a storm, of which a victim is the fortunate Ferdinand. The enchanted island. The first timid stirrings of love between Ferdinand and Miranda. Ariel. Caliban. The lovers are overwhelmed by their passion. Prospero renounces his magical powers and leaves the island. The Sea. 
On the sea comes the royal fleet, far away on the horizon, then approaching ashore slowly.
2. Royal Family
The short passage on woodwinds seems to indicate the royal family arriving, and soon turned into grant gesture, mixed with the anxiety before the storm.
The ferocious storm section is obviously a development of the open theme:
Just as his Romeo and Juliet, the love theme is given a prominent place in this work, first introduced here after the storm, and then much more developed and brought to full emotional climax, interrupted by the Ariel & Caliban themes.
5. Ariel & Caliban
Shakespeare’s play concludes on a rather pleasant note, reflected on the ending of this work; after which, the theme of the sea and royal fleet repeated, and fades away quietly.
 The Tempest, Tchaikovsky Research