That's a great question. Here is what I found out about it.  

On a stone pedestal is a bronze group of the rearing horse Bucephalus beside the standing figure of Alexander, who has his left hand stretched out to touch the shoulder of the horse. Alexander's right hand pulls at the implied reins. (The Builder of 26 April 1884 criticises the statue for this lack of reins: 'although it would have been impossible to produce the reins with which the man is supposed to restrain the horse in marble, yet such might have been done in bronze with good effect.') (1) A great swathe of drapery flows with a diagonal sweep from Alexander's left shoulder over his advanced right leg, to curl in a mass beneath the horse's rear legs. (This, too, was criticised by The Builder 'as being rather suggestive of a sheet than a garment') (2)

The group illustrates the taming, by Alexander, Prince of Macedonia, of Bucephalus, previously untamed and uncontrollable.

The head of Alexander is modelled on a bust of Alexander in 'some Florentine gallery' (3); probably the Dying Alexander in the Uffizzi, according to Rocco Lieuallan (4)

This was Steell's first major work, begun in 1829 soon after his return from Rome. It was also his last major work, as it was not cast in bronze till 1883. This was as a result of a Committee of Noblemen and Gentlemen raising funds. This committee of subscribers was formed under the convenorship of the Duke of Buccleuch. A contribution of 100 guineas came from the Town Council of Edinburgh in 1882, on condition that the statue was handed over to the Corporation on its erection (6). In 1883 a site in St Andrew Square, facing west along George Street, was granted. The statue was unveiled by the Lord Provost on 18 April 1884.

In 1916 the statue was moved from St Andrew Square to the quadrangle in front of the City Chambers, to make way for the Gladstone Memorial. The removal was undertaken by James Millar & Sons, at the cost of £130. This was paid by the Gladstone Memorial Committee.

Alexander the Great (356-323 BC), king of Macedon and founder of Alexandria. The taming of Bucephalus occurred when Alexander was a young man. According to Plutarch, when the horse was brought before Alexander's father, Alexander noticed that Bucephalus was frightened of his own shadow, so turned him to face the sun. The Scotsman of 19 April 1884 stated that Steell's intention was to display 'the predominance of mind over brute force' (5).

Inscriptions : South front of pedestal: ALEXANDER & BUCEPHALUS

East side: PRESENTED TO THE CITY / BY THE SUBSCRIBERS, 1884.

West side: MODELLED, 1832. / CAST IN BRONZE, 1883.

Signatures : On drapery beneath horse on far side from Alexander (incised letters):

JN. STEELL Sculpt

Design period : 1829-1833

Year of unveiling : 1884

Unveiling details : 18 April 1884

Information from Public Monuments and Sculpture Association (PMSA Work Ref : EDIN0187)