The Hanging Gardens of Babylon

When - 600BC
Where - Babylon (Ancient), Iraq (Modern)
Who - Babylonian king Nebuchadnezzar II
Why - A present for his wife, Queen Amytis of Media.

Temple of Artemis

When - 550BC - but do note that it was reconstructed many times, the first structure was once built in the bronze age, but the latest reconstruction in around 320BC was its largest, this remained for around 600 years, so for arguements sake i conclude this to be the real time of construction of the 'Wondrous' Temple.
Where - Epheses (Ancient) Turkey (Modern)
Who - Ephesians - 320 BC
Why - For the goddess Artemis

Mausoleum of Mausolus at Halicarnassus

When -350BC
Where - Halicarnassus (Ancient), Turkey (Modern)
Who - Artemisia.
Why - Built as a tomb for Mausolus, a ruler in Persdia, by Artemisia, his sister and wife.

Statue of Zues at Olympia

When -
Where -
Who -
Why -

The Lighthouse of Alexandria

When -270BC
Where - Alexandria, Egypt
Who - Ptolemy Soter, next ruler of Egpyt after Alexander the Great.
Why - As a symbol for this new thriving city and to of coarse guide ships into harbour from the Nile and the Mediterranean.

The Colossus of Rhodes

When -280BC
Where - Rhodes, Greece
Who - Chares of Lindos
Why - To celebrate a famous victory over the Cyprians.

The Great Pyramid of Piza

When - 2560BC
Where - Egypt
Who -Ancient Egyptians
Why - Built as a tomb for the fourth Dynasty Pharoah Khufu
And - It is important to note also that the Giza plateau iteslf is not the wonder just the main pyramid pictured below.