My artesian well just ran dry. Do I just wait for it to fill up

Typically 100psi poly is used (it's a black flexible pipe with a blue stripe). You should be using brass or poly fittings. I personally avoid nylon, just because with the threaded parts it tends to have the tool grip worn down - it's especially a pain when you go to take it off years after it's been put on, and you need to apply a lot of torque: it tends to strip the fitting, rather than loosen it. Again, make sure the fitting you're using is rated for at least 100psi.

When you connect this pipe to the fittings, you'll need to heat it up (I prefer MEP gas but propane works as well) just enough so it can go on (you still need to apply a bit of force). If you heat it too much to the point it stretches, you'll weaken the pipe and probably cause a leak - if you do this, cut the end off and start over. Use two stainless steel gear clamps to ensure a reliable connection (you can usually get away with one -- but in an outside location like a well, an extra couple gear clamps is probably cheaper than the pain of having to do this repair again).

Wrap some black electrical tape around the gear clamps afterwards, especially if you have wires nearby. This reduces the chance of the sharp edges of the clamps rubbing against the wires and causing a short eventually (I've seen it happen - remember, these pipes will move a bunch every time the pump comes on). It also makes it easier to get the clamps off later, as they won't get full of mud and insects and whatever else. Inside, unless there are wires right next to the pipes, this generally isn't needed.

Most pressure systems are adjusted so they turn on at 40psi, and off at 60psi. The pressure tank in your house buffers this so the pump doesn't cycle constantly (which is hard on pumps). You want to use higher-rated pipes and fittings because sometimes pressure switches can fail and cause the pump to stay on. Most typical domestic pumps (up to 3/4 HP) can't produce above about 90psi, though it depends on several factors including the depth of the well and length of piping.

According to the company that installed my well pump and pressure tank, they say that this thing happens. Apparently some bacteria and iron are producing stuff that is getting caught in the filter, and occasionally there is a large amount of this. Some people may have to replace their filter every week normally. As long as there's nothing that looks like sand and shells in the filter, everything should be ok.