I've been in pretty big storms before on campouts, but only two were big enough that the water would seep in through the sides of the tent. One of those times I used a tarp, rope, trees, and leaves/sticks to make a shelter and slept outside in the rain and stayed completely dry. I won't always have a tarp with me, so what are some good ways to stay dry in severe weather w/o a tarp?
Before you make any decisions, you need to decide whether it's just rain or it could have the potential to turn into a storm. If you're not sure then err on the side of a storm, just in case.
If it's a storm then before deciding on how to keep dry, the immediate action you need to take is make sure you're not at risk from being struck, and if you are minimise it. If the storm is right on top of you then crouch down, minimise your contact with the ground (i.e. don't lie down) and wait for it to pass. If it's a few miles away then see if you can get to lower ground or in a wooded area for shelter, though obviously not a single tree.
If you've mitigated the risk of lightning strike then quickly get out any waterproof material you may have and wear it as best you can - coat, hat, etc. (Such items should be accessible for this very reason.) If sensible, get yourself above the wet ground by finding or building a platform out of whatever materials might be lying around and get on top of it.
If the weather looks like it's set to stay, then look into building a shelter as best you can with the materials available. Don't forget more obvious choices as well - if you know where you are and know that civilisation is nearby for instance you might choose to call it a day, especially if you're ill prepared for the weather you're experiencing. Just make sure you know where you're going, getting lost in such weather is hardly a pleasant experience.
I am going to assume your in a wooded area not large open area and it's raining but not snowing.
First thing to do is get out of the water flow and onto higher ground
If you can build a "higher ground" for yourself (i.e. a stump) that you can stand on to get out of the water on the ground. Basically, make the water run off lower than you are.
Second thing is get the falling water away from you
If you're without a tarp or a rain jacket you need something that will repel water (I will call this material you use as a "tarpish"). I would setup a makeshift canopy out of your tarpish so that you can sit/stand under it without touching the tarpish so the water is less likely to transfer though it and run off to the ground (where you above). try and keep the tarpish tight so it wont collect water but let it run off. your first layer of tarpish is still drippy setup a second tarpish about a half foot below. obviously if you have a tarp use that first, but we are that assuming you do not.