Participants in the Cold War:

26th Century Sphere-Builders working through the 22nd century Xindi
29th Century Future Guy working through the 22nd century Suliban Cabal
29th Century Na'kuhl faction headed by Vosk across millions of years, notably through 1940s Nazi Germany
29th and 31st Century United Federation of Planets' Starfleet

A Brief Introduction to the Cold War:

The Temporal Accords were a historic agreement to preserve the integrity of the timeline. Its principles were rejected by Vosk, who started an outright rebellion using stealth time travel. Unlike Earth's historical Cold War, the Temporal Cold War involves countless unknown factions, each with their own agenda. Humanity in the 31st Century is attempting to ascertain the identity of their adversaries and to thwart their efforts.

Although the scope of the war was unprecedented, most of its casualties were unaware that they were involved in an organized conflict; hence the term "Cold War". Timelines were changed frequently, with history being rewritten or significant events being erased. Eventually the struggle broke out into full-scale war.

The Temporal Cold War came to a decisive end in 1944 New York City when Captain Jonathan Archer, Silik (the leader of the Suliban Cabal), and the crew of the Enterprise destroyed a "temporal conduit" that Vosk and his radical faction were attempting to use. In the subsequent explosion, Vosk is destroyed, and the present timeline is restored.

Major events of the Temporal Cold War:

Thousands of years ago
As a prelude to invasion, a number of planetoid-sized spheres are constructed to reconfigure space and make it suitable for transdimensional beings known as the Sphere Builders. The region becomes known as the Delphic Expanse.

Twentieth century
An unknown temporal agent assassinates Lenin, preventing Russia's Bolshevik Revolution and the rise of communism.
Vosk aids the Nazis in invading and occupying the United States during World War II.

21st Century
Jonathan Archer and T'Pol foil Xindi-Reptillian research on Humans in Detroit.

22nd Century
The Sphere-Builders tell the Xindi Council that Humans will eradicate them in the future.
The Suliban Cabal begins taking orders from the 29th Century.
Daniels infiltrates the Enterprise crew to protect them.
Enterprise is implicated in the destruction of a mining colony by the Cabal.
Enterprise, the Suliban and the Tholians struggle over a derelict ship from the future.
The Xindi test a prototype for a preemptive strike on Earth, killing 7 million.
The final version of the Xindi weapon, capable of destroying a planet, is built and then destroyed by a combination fleet consisting of Enterprise (NX-01) and several Xindi ships.

24th Century
The Federation develops the Temporal Prime Directive, a precursor to the Temporal Accords.

26th Century
The Federation defeats the Sphere-Builders at the Battle of Procyon V.

29th Century
A faction led by the mysterious Future Guy contacts the Cabal in the past to employ them for their own ends.
Vosk rejects the Temporal Accords.

31st Century
Daniels is sent back in time to protect the Enterprise NX-01.