The strong side of the offense is determined by where the tight end lines up; in run plays he'll act as an extra blocker alongside the guard and tackle on that side.
In many defenses there isn't much / any difference between free and strong safety. As far as I recall the SS will be more of a run stopper (lining up on the strong side) and the FS more of a pass coverage guy. But lots of defensive schemes don't make so much of a distinction between the two.
There are two safeties on our team, labeled FS and SS. The SS is the strong safety, and the FS is the free safety. Their exact duties depend on the defensive scheme of the team. The safeties are the last line of defense. When things get past them, it's pretty much all over. As the NFL becomes more and more a passing league, the safeties are looking and behaving more and more like additional cornerbacks on the field, and becoming more and more involved in the passing game.
The strong safety tends to be a bit larger and stronger than the free safety. A typical strong safety will weigh about 210 pounds. His job tends to be to play up closer to the line and help out in stopping the other team from running the ball. He is also responsible if players go into motion in the backfield and then go out for passes, for example the running back or fullback or h-back.
The free safety tends to be just a little smaller, perhaps 200 pounds, and just a bit faster. His job tends to be to stay back a bit, watch the play unfold, and be where ever the ball is. If there's a pass, the free safety is definitely supposed to be nearby the receiver by the time the ball gets there. Both safeties are supposed to be sure tacklers, and it's very nice if they also deliver "lights out" hits from time to time. If the offense puts a receiver in the slot, then the free safety may be called upon to cover that receiver. Sometimes instead of the safeties dividing up their jobs in terms of run support and pass support, instead the safeties will divide up the field into a left half and a right half, and each will be responsible for anything that comes into his half of the field. This type of division of responsibility is becoming more and more common, and is called a "cover-2" defense.