This question is a bit broad, but news media reported that had it not been for Covid-19, local and general elections were supposed to be held on Aug 29:
Ethiopia had been due to hold national and regional elections on Aug. 29 but postponed them indefinitely due to the coronavirus pandemic.
I don't know the exact legal basis for that date.
The timeline of the postponement/cancellation decisions isn't terribly clear. Apparently there were multiple events:
Back in March, Prime Minister Abiy Ahmed postponed parliamentary elections scheduled for 29 of August. Then, a couple months later, the government decided to postpone the election until 2021.
According to a somewhat more detailed account by Al-Jazeera in June:
Ethiopia’s parliament has approved allowing Prime Minister Abiy Ahmed to stay in office beyond his mandate after elections planned for August were postponed due to the coronavirus pandemic.
The vote on Wednesday – 114 in favour, four against and one abstention – came two days after a leading opposition politician resigned as speaker in an apparent protest against the decision to delay the election. [...]
Lawmakers did not specify when the new elections would happen, however, their vote was an endorsement of recommendations by the Council of Constitutional Inquiry, an advisory body that had held public meetings to decide a way forward after the delay.
The body recommended for the “elections to be held nine to 12 months after the coronavirus is deemed not to be a public health concern”.
Ethiopia’s election board announced in March that it would be impossible to organise the vote on time because of the pandemic, in which 2,506 infections have been confirmed in the country with 35 deaths.
The circumstances meant that the election could not happen before legislators’ terms end in October.
The Ethiopian constitution does not clearly address the path forward in the unusual situation.
N.B. According to CGTN that was the upper house making that decision; also the speaker who resigned was from the TLPF. (Alas most Western media has provided poor coverage of this crisis in Ethiopia before the fighting broke out; the most detailed account I found are from Arab or Chinese media...)
The vote on the resolution came two days after Keria Ibrahim, a top official of the Tigray People's Liberation Front (TPLF), resigned as speaker of its upper house in apparent protest at the delay.
Keria accused Abiy's government of taking away Ethiopians' sovereign rights in a televised speech, but she fell short of elaborating the accusation.
"I can't be an accomplice when the constitution is being violated and a dictatorial government is being formed," she said. "I have resigned not to be collaborator (with) such a historical mistake."
Soon after the Tigray elections were held in defiance of Abiy's cancellation order
Ethiopia’s upper house of parliament, which mediates constitutional disputes, ruled that the polls for regional parliaments and other positions were unconstitutional.
As for the broader context, the TPLF was unhappy with Aiby's attempt to restructure the ruling coalition into a single party. As the BBC notes:
Although Tigray represents just 6% of Ethiopia's population of more than 100 million, the TPLF used to be the dominant force in Ethiopia's ruling coalition but its power has waned since Mr Abiy became prime minister.
Last year, he dissolved the ruling coalition, made up of several ethnically-based regional parties, and merged them into a single, national party, the Prosperity Party, which the TPLF refused to join.
And yes, the cancellation/postponement of national election by Abiy provided a political opportunity to challenge his legitimacy:
[TPLF leaders say] they have been unfairly targeted by purges and allegations of corruption, and say Mr Abiy is an illegitimate leader, because his mandate ran out when he postponed elections due to coronavirus.