Mrs Leadsom said she "no longer believed" that the government's Brexit approach "could deliver on the referendum result".
In a letter written to Mrs May, Mrs Leadsom said she did not believe that the UK will be "truly sovereign" through the deal proposed, and said a second referendum would be "dangerously divisive".
She also said there had been "such a breakdown of government processes that recent Brexit-related legislative proposals have not been properly scrutinised or approved by Cabinet members".
A former Tory leadership candidate in 2016, Mrs Leadsom was a high-profile Brexiteer, becoming a figurehead of the Vote Leave campaign.
She was one of a number of Brexit-supporting colleagues in the so-called Pizza Club who were absent for the start of Prime Minister's Questions on Wednesday.
Posting the letter on Twitter, she wrote: "It is with great regret and a heavy heart that I have decided to resign from the Government."
She said that she was "proud" to have served in Theresa May's Government since 2016 and had stayed in Cabinet "to shape and fight for Brexit" but there had been some "uncomfortable compromises along the way".
She added: "The tolerance to those in Cabinet who have advocated policies contrary to the Government's position has led to a complete breakdown of collective responsibility."
The night before an election the government is in total freefall. Brexit caused this crisis, only stopping it will provide a way out. We need a referendum and a general election ASAP. https://t.co/XQzQaadWnf— David Lammy (@DavidLammy)22 May 2019
Mrs Leadsom also addressed the timing of her resignation, which falls on the eve of polling day in the European elections.
She said she had decided to step down ahead of Thursday's Business Statement to the Commons because she felt she could not announce a Bill with "new elements that I fundamentally oppose".
🚶♀️ As Westminster descends deeper into chaos, Andrea Leadsom is the 42nd minister to depart from Theresa May's government.
👀 This is a far cry from "strong and stable".https://t.co/I4iuhJfhDG— The SNP (@theSNP)May 22, 2019
She added: "I fully respect the integrity, resolution and determination that you have shown during your time as Prime Minister.
"No one has wanted you to succeed more than I have, but I do now urge you to make the right decisions in the interests of the country, this Government and our party."
Her resignation, which comes amid a backlash over Theresa May's new Brexit plan, is the 36th by a minister under Mrs May, 21 of whom have been over Brexit.
Mrs May said she was "sorry" to receive Mrs Leadsom's resignation letter, writing in reply that she was "grateful for the support you have given over the last three years" in working to deliver Brexit.
But the Prime Minister said she disagreed with the assessment Mrs Leadsom gave about the Government's approach now.
She said: "I do not agree with you that the deal which we have negotiated with the European Union means that the United Kingdom will not become a sovereign country.
"If the deal is passed, the UK will leave the European Union. We will leave its Common Agricultural Policy and its Common Fisheries Policy.
"We will end the jurisdiction of the European Court of Justice and make our own laws in Westminster and our devolved parliaments.
"We will stop sending vast annual sums of taxpayers' money to the EU, and spend it on our priorities instead - such as our National Health Service and our children's schools."
Mrs May said she agreed that a second referendum would be divisive, but said the Government was not proposing to hold one.
She wrote: "I continue to believe that the arguments against a second referendum are strong and compelling, and will continue to oppose one."
The PM said she also did not recognise what Mrs Leadsom said about decision-making in Government.
"Indeed, through your own work chairing the Parliamentary Business and Legislation committee of Cabinet, you have been an important part of ensuring that Brexit-related legislative proposals are properly scrutinised, and the whole Cabinet have listened to the assessments you have given at Cabinet meetings," Mrs May said.
"I have valued the frank and productive discussions we have had over the last three years.
"As you say, there are important elections tomorrow, and many Conservatives are working in a challenging environment to support our excellent candidates. I am sorry to lose someone of your passion, drive and sincerity from HM Government in this way."
Mrs Leadsom reportedly rang the PM to tell her she was quitting 30 minutes before she made it public, according to The Sun's Tom Newton Dunn.
The Prime Minister will meet Sir Graham Brady, the representative of Tory backbench MPs, on Friday for a meeting which could seal her fate.
A Downing Street spokesman said: "Andrea Leadsom has served with distinction and great ability as a member of the Government, and the Prime Minister is grateful for all of her work.
"We are disappointed that she has chosen to resign, and the Prime Minister remains focused on delivering the Brexit people voted for."
Mrs Leadsom worked in finance before becoming an MP in 2006. She represents South Northamptonshire.
She was appointed Economic Secretary to the Treasury in 2014, and was later moved to the Department of Energy and Climate Change under Secretary of State Amber Rudd.
In 2016, she stood to become leader of the Conservative Party against Mrs May, but came second in the first two ballots before withdrawing her bid.
She was criticised for comments made during the race about being a mother when Mrs May was not, forcing her to apologise for being "guilty of naivety."
Following a cabinet reshuffle in 2016, Mrs Leadsom was made Secretary of State for Environment, Food and Rural Affairs.
She was appointed Leader of the House of Commons the following year.
Commenting on Andrea Leadsom's resignation from the Cabinet, Labour Party chairman Ian Lavery said: "The Prime Minister's authority is shot and her time is up.
"While the Tories are ripping themselves apart, our country is in crisis. For the sake of the country, Theresa May needs to go, and we need an immediate general election."
Labour's deputy leader Tom Watson called it a "slap in the face" for her Tory colleagues ahead of Thursday's elections.
He tweeted: "I accept that she may want to go but to do it the night before an election looks odd. It's also a slap in the face to Tory party members who are working hard to get their candidates elected tomorrow."
Fellow Conservative Party MPs also described Mrs Leadsom's decision as "absolutely the right thing to do".
Scottish Tory Ross Thomson tweeted: "Sad to see @andrealeadsom leave Government but absolutely the right thing to do. This new Agreement breaks the last promise that was possible to break on a second EU referendum."
Another Conservative MP, Chris Heaton-Harris, tweeted: "A great shame, but completely understandable.
"Tomorrow she would have had to announce a Bill containing elements (2nd referendum and Customs Union) that she simply could not support in good faith."
Conservative MP Craig Tracey appeared to mock the situation as Theresa May lost a key member of her Cabinet.
"Just heard Larry the Downing Street Cat is considering his position," he tweeted.