This question is bound to get a lot of people going as both of your alternatives carry some of the truth of the situation. The molecules in the air are all pulled towards the centre of the Earth but do not all fall down because of their thermal motion and the resulting collisions. At the bottom of a column of air (i.e. on the ground) the pressure is die to the adjacent molecules continually hitting the ground. Those molecules, in turn, are collided with by molecules higher up and so on, up tot the edge of the atmosphere ( it would go on for ever, in fact, getting less and less dense in an approximately exponential way). Every molecule in a vertical column will be imposing its weight on the molecules below it so you could say that the pressure is actually caused by the weight of all the molecules overhead. If Earth were cold enough, there would be a 'thin' layer of liquified air all over the surface and the pressure would be easier to appreciate and to calculate but there is no other force available to increase the pressure when it's gaseous so you can say it's still due to weight. Just one caveat and that is that the weight of the molecules at the very limit of the atmosphere is just a bit less (not a lot, of course) than at the surface because they are a couple of hundred km further away.
