The current stage of the war began in March 2015 when Saudis launched a direct military intervention to return Hadi to power in Sanaa.
Hadi had been deposed three months earlier by a Houthi uprising and the armed forces which had remained loyal to the previous president Saleh who had been guided out of power by his erstwhile backer, the US, as part of the Arab Spring.
Houthis had boycotted the 2012 single-candidate “election” which confirmed Hadi as president, and had been in a slow-motion revolt against him since 2014 and had taken control of the capital Sanaa in September 2014.
— When the Saudi intervention begun the Houthis (Zaidi Islamists) and Saleh’s military (secular) were in control of Sanaa and Aden (the other large city) both and in a more powerful position. Aden was retaken by Hadi remnants with Saudi support a few months later in June 2015.
— After the Saudi intervention started the Houthis were actually able to take and for a while hold a small piece of territory inside Saudi Arabia. (Territory settled by tribes centered in Yemen.)
— Southern Movement/The Southern-Transitional Council are southern secular separatists who ideally want the old seperate South Yemen with the capital in Aden back. They’re backed by United Arab Emirates and loosely aligned with Hadi against the Houthis (primarily because that’s what UAE wants of them).
The present lines of control: