May says no to EU proposed deal on the future UK and EU economic relationship and Backstop for Northern Ireland. The EU’s offer to the UK consisted of:
1. UK staying in the European Area and Custom Union. This would mean the UK obeying EU rules and not being able no establish separate trade deals with the rest of the world.
2. Free trade deal that would include Northern Ireland remaining within the custom union to avoid a hard border with the Republic of Ireland (see Good Friday Agreement for more details).
May rejected point 1 because it would fail to deliver on the referendum by staying within European control and without voting rights. Parliament rejected point 2 because it threatens the integrity of the UK union.
May wants from the EU an agreement that sees frictionless free trade of goods that protects the integrity of the UK union in the same way the EU wants to protect its own Union.