Ok.
Depending on the offensive strengths of the team, and opposing team's defensive strengths these two factors sort of dictate how a team starts a drive/approaches moving downfield.
Field position and time remaining also play a factor.
Typically a passing down is a 3rd and long (10+ yards to gain for a first down), or 2nd and long (depending on how long, and field position).
In the waining minutes of a game a team may elect to run the ball on a draw or HB delay to keep the clock running and/or maybe sneak out a first down if they've got a good RB and run-blocking O-Line. Typically though running the ball on 3rd and Long is done by the team in the lead...to keep the clock running.
Ex. 2008 Minnesota Vikings had the best run-D in the league keeping opponents to a 76.9 average yds/gm. Their passing was middle of the pack @ allowing 215 yds/gm. So the way to beat them was to pass the ball...however they also have one of the best pass rushers in the league (Jared Allen) so if your OT wasn't up to task Minny could wreak havoc on your passing game given they've got at least one very good cornerback in Antione Winfield and DTs that allow the MLB/FS to get free on a blitz.