How is a new U.S. Supreme Court Justice appointed?

After the death of U.S. Supreme Court Justice Ruth Bader Ginsburg over the weekend, the big question has been whether President Trump will have the time and support in the Senate to appoint another Judge to the Supreme Court.
This a big topic, so we'll attack it in three parts:
How is a new Supreme Court Justice appointed?
Why does the appointment of a Judge matter to Americans?
Who are Trump's leading nominees for the Supreme Court?
The nomination process
President Trump will nominate a candidate - that nomination is sent to the U.S. Senate Judiciary Committee.
The Senate Judiciary Committee will gather information on the nominee - this normally takes about a month.
The Senate Judiciary Committee will hold a hearing of the nominee - this is where witnesses (both in favour of and opposing the nomination) will speak.
The Senate Judiciary Committee then takes a vote and sends their recommendation to the Senate. The options are:
that it be confirmed
that it be rejected
that there is no recommendation
The full Senate will debate the nomination. 51 of 100 Senators need to vote to end the debate.
The Senate will vote on the nomination. 51 of 100 Senators need to vote for the nominee for them to be nominated to the bench.
Can Trump’s nominee get (realistically) knocked back by the Senate?
Yes - but 4 Republicans must vote against a new nominee, as well as all Democrats and both Independents.
If only 3 Republicans vote against the new nominee, the deciding vote will fall to Vice President Mike Pence, who will vote with the President for the nominee.