Surgical airway is indicated when there is no appropriate alternative means of oxygenating or ventilating a patient.
Such situations include:

Failed bag mask ventilation, failed extraglottic airway, and failed tracheal intubation;

Limited access to the upper airway due to entrapment;

Unable to attempt airway patency manoeuvres via nose or mouth due to massive facial trauma or burns;

Open wound of anterior neck exposing transected trachea.

In the latter three cases it is often appropriate for the surgical airway to be the initial airway attempt.
Terminology is important. Surgical airways, when effectively created, are not ‘failed’ airways but successful, alternative airways. Adopting this terminology creates a mindset in which the airway will be performed as soon as it is indicated, maintaining oxygenation, rather than delayed to the last possible moment amidst plummeting oxygen saturations, for fear of being considered a failure.
A surgical airway requires a scalpel incision through the cricothyroid membrane followed by insertion of a tracheal tube. Many practitioners are more comfortable with the idea of a needle cricothyroidotomy, in which oxygenation is attempted through a narrow cannula. However needle cricothyroidotomy is associated with a significant failure rate. The UK National Audit of Emergency Airway Management, NAP4, showed a 60% failure rate for a needle technique, whereas surgical techniques were ‘almost universally successful’.
The amount of blood that appears when the neck is cut renders visualisation of the structures impossible. Therefore a tactile approach is recommended, in which a finger is inserted through a cricothyroid membrane incision into the trachea. The tracheal lumen and rings can be palpated, and then a bougie placed alongside the finger. The finger is then removed and a tracheal tube or tracheostomy tube is then railroaded over the bougie into the trachea, the cuff inflated, and waveform capnography connected to confirm intratracheal placement.
This video demonstrates the recommended scalpel-finger-bougie method (courtesy of SydneyHEMS.com):

This video demonstrates the method on a human cadaver (courtesy of EMCrit.org):

This video demonstrates the method on a live patient (courtesy of EMCrit.org):

Further reading: Paix BR, Griggs WM. Emergency surgical cricothyroidotomy: 24 successful cases leading to a simple ‘scalpel-finger-tube’ method. Emerg Med Australas. 2012 Feb;24(1):23-30