What it is?

A Scott connected transformer is a type of circuit device used to convert a three-phase supply (3-φ, 120-degree phase rotation) to a two-phase (2-φ, 90-degree phase rotation), or vice-versa. 
The Scott connection evenly distributes a balanced load between the phases of the source.

How it all started?

In the later part of 19th century, two phase generators were used to drive three phase motors. for this Edison’s rotary convertors were used. As the usage of rotary convertors were expensive, an engineer from Westing House Charles.F.Scott designed a transformer that converts three phase to two phase or vice versa. Hence the transformer is named after him.
Now as most of the equipment runs on either single phase or three phase, Scott connected transformers became obsolete. However still in some places like the electrical locomotive stations, these transformers have relevance.

What is Scott Connection?

The Scott connection uses two single-phase transformers to transform three phase voltages and currents into two-phase voltages and currents.

The first single phase transformer is called the ‘‘main transformer’’. It has a center-tapped primary winding connected to the three-phase circuit with the secondary winding connected to the two-phase circuit. The ends of the center-tapped main primary winding are connected to two of the phases of the three-phase circuit. You may observe in the below left side figure R and Y terminals of a three phase supply are connected to the two terminals of main transformer.

The second single phase transformer is called the ‘‘teaser transformer’’. It has one end of its primary winding connected to the third phase of the three-phase circuit and the other end connected to the center tap of the primary winding of the main. The Scott connection requires no primary neutral connection, so zero-sequence currents are blocked.
You may observe one terminal of the teaser transformer is connected to the center tap of main winding and the other terminal to B phase. The term mentioned as Scott tap is explained further in this article. In order to neutralize the ampere turns of the center tapped primary winding of the main transformer, the two halves are wound on the same core leg. Also the center tapped winding of the main transformer may be interleaved to cancel the magnetic effect of current entering at the center point.
The secondary windings of both the main and teaser transformers are connected to the two-phase circuit.

If the main transformer has a turns ratio of 1: 1, then the teaser transformer requires a turns ratio of 0.866:1 for balanced operation. The apparent power supplied by the main transformer is greater than the apparent power supplied by the teaser transformer.

How to verify?

You may find many sites or blogs or texts that explain the fundamentals with lots of phasor terms. In order to make it simple and understandable, the following may be performed. Take two single phase transformers of same capacity. You name the first transformer as “Main Transformer”. The main transformer shall have a center tap on its primary winding as explained above. The center tap shall be connected to the starting terminal of the second transformer called as “Teaser transformer”.  
In the second transformer you take a tap at 86.6% of the primary winding. This tap is called the Scott-tap.  
Now switch on the transformer with three phase supply and observe the primary currents and voltages.  You will observe that the primary currents in both transformers have the same magnitude and the rated current is passed when the primary voltage of main transformer is V and that of teaser transformer is 86.6% of V.
The teaser transforms only 86.6% of the apparent power transformed by the main. The total apparent power transformed by both transformers is greater than the total apparent power delivered to the two-phase load. 

We also observe that the total real power delivered to the two phase load is equal to the total real power supplied from the three-phase system.

How to Construct? 

There are two methods of constructing Scott-Connected transformers. 
The first one is make two single phase transformers inter connect as explained and lower them in single tank or tank the two transformers i.e. main and teaser in two separate tanks and connect externally. 
The second one is make a three limbed core (similar to that of a three phase transformer). Lower the main and teaser transformers on the two extreme core limbs. this makes the center limb without any winding. However in order to facilitate the return path for the flux the area of the center limb should be increase by SQRT2 or in layman terms the center limb area should be increased by 41% . Otherwise there would be localized heating at the center limb because of high flux density. For example if the working flux density of the main and teaser transformers is 1.65T then in the center limb the flux density would be increased by 41% i.e. it would be 1.65 x 1.41 = 2.327T which is more than the saturation point for any electrical grade steel. Due care shall be taken in order to keep the flux density well within the saturation point under any extreme system conditions.  

General 

Most Scott Connected transformers are designed with two separate two limbed cores but the introduction of automatic core cutting machines made it more efficient to utilize a three phase core. however as explained above, this has to be over sized it is relatively a more expensive method.