Golf is an outdoor sport (typically), and that means its players and competitions they play in are susceptible to the elements and weather. Add in that golfers play with equipment made from metal, and there's an additional safety consideration.
The PGA Tour is well aware of the weather forecast for their tournaments, and they often set up the tee times and groupings -- as well as playing windows and the number of starting tees used -- with advance knowledge of the weather. If they know weather conditions are going to be so bad that it would be foolish or dangerous to play golf, then they're going to delay the start of play or cancel play for a whole day altogether.
However, there are particular circumstances under which the PGA Tour will suspend, delay or cancel play during a tournament day of a PGA Tour event.
The PGA Tour will halt play when there is lightning reported within 5 miles of a tournament venue, clearing the course of players and spectators alike. Lightning is dangerous and can strikes golf clubs, as well as trees that players and spectators might hide under for safety. The Tour wants to prevent any potential for injury.
The Tour will also stop play if it knows the wind is so strong that a golf ball will not remain still when a player is preparing to hit it, particularly when putting on the greens. The golf ball staying still is a key factor in the game. Though the rules have changed to no longer penalize players whose balls incidentally move because of the wind while their club is behind the ball, players will not play if they cannot expect the ball will remain still.
The Tour will also suspend or even cancel play if the course is deemed unplayable. A course can be deemed unplayable for a variety of reasons, including if it is so saturated with rain that water is puddling up on greens or streams of water are flowing on fairways. Basically, if surfaces that are typically just grass (or sand bunkers) have water covering them instead, then play needs to be stopped so the water can be cleared.
Sometimes PGA Tour weather delays are short, less than a half-hour. Sometimes they are full-day cancellations, leading to a round or several rounds of a tournament to be pushed forward by a day. Under PGA Tour regulations, almost every event on the schedule cannot be completed after Monday (except for The Players Championship and in other special circumstances), so delays and cancellations do have a tremendous impact on a golf tournament's outcome.