What appear to be curved lines on a map are actually straight lines when viewed from the top (plan view) on globe.
Great Circle Navigation: Commercial aircraft by and large fly in a straight line called great circle. Great circle is the shortest distance between any two points on earth. Aircraft fly along rhumb line (Curved path) if there are no ground based radio navigation aids and also when it is not equipped with Global Position System (GPS) or Inertial Navigation System (INS).
A Rhumb line path, flown with the help of compass, cuts all meridians (lines joining North pole and South pole) at equal angles, therefore it is a curved line and not a straight line.
Terrestrial Based Navigation System: Terrestrial based Navigation Systems such as VORs (Very high frequency Omni directional Range) enable aircraft to move in a straight line from point A to B, but their range is limited to about 200 miles due to line of site problem. Hence aircraft flying across oceans or mountain valleys can not be provided direct communication, navigation and surveillance services using terrestrial based systems (see VOR pic above).
Satellite Based Navigation: Satellites have solved ‘line of site’ problem because of their height above ground. Satellite based Communication, Navigation and Surveillance (CNS) systems can have a range in excess of 1000 miles. It also solves the line of site problem in valleys. Satellite based CNS ATM (Air Traffic Management) systems such as Automatic Dependent Surveillance (Contract) commonly known as ADS (c) has overcome Communication and Surveillance problem both on oceans and in valleys. Ground controllers can now communicate with pilots through CPDLC (Controller Pilot Data Link Communication) and ACARS (Aircraft Communication Addressing and Reporting System) over a thousand miles away. This system is functional for last 30 years or so across the globe except those jurisdictions who control small airspace over plain land (see pic above).
Flexible Route Planning: Satellite based CNS ATM systems offer airlines the flexibility to change route on the shortest notice. Online information about wind speed and direction prevailing at various altitudes enable aircraft to avoid strong head winds or take advantage of the tail winds (see above map).
Polar Routes: Upper winds—at higher altitudes move from West to East, due to earth rotation from West to East. In winters winds are stronger than summers. Their speed can range between 60 to 100 miles an hour. Depending on the direction of upper winds ground speed of an aircraft would increase or decrease proportionately. Aircraft flying from Asia to North America in a ‘north south’ direction (cross wind) through north pole save upto 30 minutes of flying time. On their way back they may be flying in an easterly direction to take advantage of tail winds (see above map).
Zig Zag Routes: Certain jurisdictions do not allow aircraft to fly through airspace reserved for military training. The aircraft, therefore, circumnavigate prohibited areas and danger areas (see above map).
Extended Twin Engine Operations (ETOPS): Most of the long range aircraft (B777, A330, B787, A350 etc) are now a days equipped with two engines rather than three or four. This saves fuel and maintenance costs. However, while flying, these aircraft must remain within specified range of an airport where it can land in case of one engine failure. This limitation does not affect them significantly because ETOPS allowed to modern aircraft can be upto 3 hours hours of flying time from alternate airports. Hence practically these aircraft fly straight routes all the time.