Experienced trial lawyers will often tell you that "jury selection" is misnamed—it's actually jury deselection.
Different jurisdictions use different methods. In a Texas non-capital felony trial, the jury panel is ordinarily sixty people. Each juror is assigned a number. The court brings the panel in to the courtroom and lines them up on the benches in numerical order—one through twelve on the first row, for example, thirteen through twenty-four on the second row, and so forth. (Lawyers often refer to the jurors by number instead of name because numbers are easier to keep track of.)
The judge talks to the jurors first. Some judges will talk for a few minutes and leave the lifting to the lawyers, and some judges will talk for hours. Usually the judge will introduce the jurors to general principles—the presumption of innocence, the right to remain silent, the burden of proof—and establish that all of the jurors can follow the law.
After the judge, the prosecutor takes her turn. She will usually talk about the elements of the case and the legal principles that she thinks are important to the case. If, as in many Texas felony cases, the defendant has chosen to have the jury set punishment, she will commit the jurors to considering the top end of the punishment range. She'll also ask questions intended to find the people who might be prejudiced against the State or the applicable law.
After the prosecutor, the defense lawyer takes his turn. He might talk about the legal principles that he thinks are important, commit the jurors to considering the bottom end of the punishment range, and seek out people prejudiced against his client or his defense.
In most non-capital cases all of this questioning is done in the presence of the entire panel. The goal is not to choose twelve jurors as much as to find and eliminate the least favorable jurors and indoctrinate those who remain. Potential jurors familiar with this heuristic can hack jury selection by talking a lot if they don't want to serve, or keeping their mouths shut if they do.
The lawyers will not usually talk to the jury about the facts of the case. While talking about the facts of the case is not legally prohibited, most lawyers think it is.
One big no-no in jury selection is improper commitment questions. A commitment question is a question that asks a juror to commit to doing or not doing something ("if the State doesn't prove its case beyond a reasonable doubt, will you acquit?"). An improper commitment question is a question that asks a juror to commit to doing or not doing something that the juror is not required to do or not do ("if you learn X, will that create a reasonable doubt?") or that includes unnecessary facts ("will you consider probation if the defendant is gainfully employed?"). The facts-of-the-case hobgoblin probably arises from a misunderstanding of the rule against improper commitment questions.
After the parties talk to the jurors, the lawyers approach the judge, out of the jury's hearing, with their challenges for cause. A challenge for cause is a reason to remove ("strike") a potential juror who for some reason cannot follow the law. The prosecutor will read the judge her list of challenges for cause ("Number two—can't consider full range; Number six—can't follow the law; Number twelve—can't judge; Number fifteen—one-witness rule" and so forth), the defense will agree or object to each challenge. Then the defense will read the judge his list of challenges for cause and the State will agree or object. It is during this phase that jurors will commonly be called up to the bench so that the judge can resolve challenges for cause. The judge will rule on each contested challenge for cause.
The judge and the lawyers then check and doublecheck which jurors have been challenged for cause, and figure out what the "strike zone" is. The strike zone is the group of jurors who might make it onto the jury—the first thirty-two who haven't been struck for cause.
Why thirty-two? Because after the challenges for cause, the lawyers retire to their corners and mark off their peremptory challenges. Peremptory challenges are strikes for any reason (other than race or sex; challenges based on race or sex are constitutionally forbidden). Each lawyer can use ten peremptory challenges in a Texas non-capital felony case. Twelve plus twenty equals thirty-two.
So the lawyers separately mark their peremptory challenges on their jury lists, and then turn the lists in to the clerk, who collates the strikes and makes a list of the first twelve unstruck jurors. The lawyers double-check the list (to make sure that no juror who was struck is on the list) and the judge calls the names of the twelve jurors and excuses the rest with the thanks of the court and work excuses.
(Special rule: alternate jurors. If there are going to be alternate jurors, the strike zone for the alternates includes the first three or four (if one or two alternate jurors are to be chosen) or seven or eight (if three or four alternate jurors) unstruck jurors past the strike zone. Each side can use one additional strike (if one or two alternate jurors) or two additional strikes (if three or four alternate jurors) in this alternate strike zone.)