Unlike sane countries if the US government doesn't pass a budget in time then federal workers can go unpaid. Many federal workers are nevertheless still forced to work (via summons) despite not being paid. This particular shutdown is due to a disagreement about providing funding for a border wall between the US and Mexico (one of Trump's campaign promises, though he said "Mexico will pay for it").

Keep in mind that the shutdown began during the previous congress, when Republicans controlled the House, Senate, and the Presidency. It has extended to the current congress where the Democrats now control the house.

To break down some additional bits and pieces for folks about how the US government works:

A bill (such as a budget) needs to be passed by both the House of Congress and by the Senate and also approved by the President.

Senate rules make it possible to infinitely delay (effectively block) any bill without support of 60% of the Senate (so even a bill that would pass the Senate may not even be voted on if more than 40% of the Senate opposes it).

If the President vetos (doesn't sign) a bill the house and senate can only override that with a 2/3 majority vote in both.