Equestria is not ruled by Princess Celestia and Princess Luna as an absolute monarchy.  These two princesses share political power with an Elected Pony Senate which is elected every seven years at least, but the ruling princesses can call for an early Senate election as their discretion. 

Elected Pony Senate

The Senate meets in Canterlot once a year for 30 days.  It is composed of 100 elected senators (30 Earth Ponies, 30 Pegasus Ponies, 30 Unicorn Ponies, and 10 senators representing non-Pony races).

Legislative Authority and Royal Assent

The ruling princesses cannot pass new laws without the consent of a Senate majority.  On the other hand, a Princess must grant her personal royal assent to bills passed by the senators

Authority of Princesses Celestia and Luna

When Princess Celestia is present in Canterlot when the Senate is in session, she as senior princess has veto power over bills passed by the Senate.  However, if Celestia is out of town at the moment, Princess Luna has the authority to grant assent to or veto Senate bills. 

On the other hand, Celestia, as senior Princess, alone has the authority to call for an early Senate election.

Development of Parliamentary Democracy

Technically, Princess Celestia, as senior Princess, has the authority to appoint governmental ministers (Princess Luna, when she was turned from evil and regained her royal authority, had gained the power to formally appoint ministers when Celestia was incapacitated or absent from Canterlot).  However, it has become unwritten convention that ministers should be members of the Elected Pony Senate and supported by a Senate majority.

Over the years and centuries, the Senate has developed two main parties:  the Sun Party and the Moon Party.  Ideologically, there is little difference between the two parties.  The leader of the party with a Senate majority does preside over the ministers as prime minister, though.

The Sun Party gains much of it's support from Pegasus Ponies while the Moon Party is solidly supported by Earth Ponies.  Unicorn Ponies and non-Ponies tend to be swing voters.

Autonomy of the Crystal Empire and the Friendship Rainbow Empire Within Equestria

Much like the devolved governments of Scotland, Wales, and Northern Ireland within the British United Kingdom, the Crystal Empire and the Friendship Rainbow Kingdom have considerable local self-government within the larger kingdom of Equestria. 

The Crystal Empire basically is a city-state jointly ruled by Princess Cadance and Prince Shining Armor.  They can issue decrees within the Crystal Empire city-state having the force of law.  However, the local pony citizens can appeal to Princess Celestia, Princess Luna, or the Elected Pony Senate in Canterlot to veto local laws.

Princess Twilight Sparkle governs her own Friendship Rainbow Kingdom with Applejack, Rarity, Rainbow Dash, Fluttershy, and Pinkie Pie serving as her viceroys.  The capital is Ponyville.  This kingdom extends up to a 20-mile radius around Ponyville.   Princess Twilight must act on the advice of her viceroys when making local decrees, and local citizens can appeal to Princess Celestia, Princess Luna, or the Elected Pony Senate to overturn local laws.