2013-32 Shutdown indeed.

As there was no last moment agreement on funding the U.S. Federal Government, the shutdown started.

What happened?

• The budget has not been approved on time 

How it unfolded? 

• The New York Times provides a useful graphic 

How will it work (or rather - not work)?

• The US government will continue to operate; but only with essential workers
• Non-essential workers will be furloughed. What is a furlough? It is a temporary unpaid leave leave caused by demands of the employer. Most other workers may be unpaid, for the duration of the shutdown, but will be paid afterwards.
• Military personnel will be paid.

The New York Times provides another useful graphics 

Mechanism:

US has an unusual system of government in that the legislative and executive branches are separately elected. In Canada, certain legislation (including on the budget, as is now the case in the US) are automatically votes of confidence in the government. If such a vote fails, the government resigns and a new election is held. Parliamentary majority is usually sufficient, because of party discipline; under a minority government, the governing party needs support from other party or parties to pass legislation.

How is US budget resolution approved?

The House and the Senate need to agree on the resolution, and the president needs to sign it. If the president vetoes the resolution, the House and Senate can override it with a 2/3 of the vote.

In this case, the House and the Senate did not agree.

Why the shutdown?

The Republicans, who have majority in the House of Representatives, tried to impose conditions on the budget resolution. The Senate rejected the conditions.

While one of the conditions involved approval of the Keystone XL pipeline from Alberta to the Gulf-area refineries, the main issue is the Affordable Care Act, or Obamacare. The Republicans are dead set against it; the House held around 43 votes rejecting it.

More details are here

So who is right?

This is politics, but with economic consequences. Here is a comment from robertreich.org:

"[...] the only settled way we know what the American people want is through the democratic process. And the Affordable Care Act (Obamacare) is the law of the land. A majority of the House and Senate voted for it, the President signed it into law, its constitutionality has been upheld by the Supreme Court, and a majority  of Americans reelected the President after an election battle in which the Affordable Care Act was a central issue."

So this is about the Republican party achieving goals by means beyond the normal constitutional system.