Who are the three Muslim groups fighting in Iraq?

It depends on what you mean by "groups".

Broadly, the three ethno-religious groups in conflict in Iraq are the Sunni Arabs, Shiite Arabs (the majority), and the Kurds (who are a distinct ethnic group - but most are Sunnis religiously).

However, if the Word "group" is intended to refer to the actual armies or militant organizations, there are a vast number of different groups fighting in Iraq, not just three. However, the three most prominent (and which all others are allied - to one or another) are:

1. Government of Iraq: Prime Minister Nouri al-Maliki is the leader of the Iraqi Government in Baghdad. His government is the only one of the three which has true recognition internationally and is seen as the legitimate leadership of Iraq. However, within Iraq, al-Maliki and his government are seen as Shiite sectarians who promote the interests of Shiite Arab Iraqis over those of Kurds and, especially, Sunni Arab Iraqis. The feeling of marginalization by Iraq's minorities is growing as al-Maliki has made the additional steps of inviting Iranian military assistance and allowing for the inclusion of Shiite militias as allies. The US is strongly considering assisting the Government of Iraq with airstrikes against ISIL. Kurds are nominally loyal to the Iraqi Government and have not militarily engaged with Iraqi national forces. The Iraqi Army is seen as unwilling to vigorously defend Iraq by both ISIL and the Kurds.

2. Islamic State of Iraq and the Levant (ISIL):ISIL is an extremist Sunni Islamist organization which used to have links to al-Qaeda before al-Qaeda condemned them for their barbarity. They are fighting to bring the Sunni majority into power, especially in Sunni majority regions of both Syria and Iraq, and impose a more fundamentalist view on politics. They consider themselves to be fighting the injustices and violence of the current Assad regime and the inequalities of the US-supported Government of Iraq. ISIL is primarily supported and armed by Saudi Arabia, but the US and UK have indirectly given ISIL millions of dollars and equipment since US and UK military vehicles and assets left to the Government of Iraq have fallen into ISIL hands (especially with the fall of Mosul).

3. Iraqi Kurdistan and the Peshmerga (Kurds):The Kurds, a Sunni ethnic minority (as opposed to a religious minority), are primarily aligned with the Government of Iraq, but are overall more concerned with protecting their minority in northern Iraq. Iraqi Kurdistan is the name of their autonomous democratically-elected governing council and the Peshmerga is the name of the militias that fight on behalf of that authority. Peshmerga forces, unlike the Iraqi Army, are seen to be incredibly loyal and ISIL has avoided attacking positions held by the Peshmerga for this reason.