Why does bathwater make a vortex in the plughole? When I google for this I just get stuff about whether or not the Coriolis effect makes it go clockwise or anticlockwise, but I don't care which direction it turns in. I want to know why it turns at all. Why doesn't the water just fall straight down the hole?
One wrong theory is that air wants to return up the pipe, but that only applies if there's a closed vessel like a wine bottle at one end of the pipe or the other. My bath drain empties into the back garden and the air can return via the bathroom window.
Another is that pre-existing rotational currents in the bath get amplified as the water is drawn to the plughole just like a ballerina pulling her arms in, but there are two problems with this. Firstly, under this theory you'd expect the speed of the vortex at the plughole to be proportional to the speed of the pre-existing currents, but experience suggests that every plughole has its own favourite speed that depends on its geometry, the water depth, etc, so that you get pretty much the same speed whether you stir the bath a little or a lot. Secondly, by leaving the tap running (taking care not to inject angular momentum) we can keep the vortex going forever, but if the only driver was the pre-existing currents, then surely they'd be depleted by viscous friction and falling down the plughole. Some other effect must be driving the vortex continuously. There's energy available from the loss of gravitational potential energy of the water, but why does it get turned sideways to make this vortex?  
 A: Do this at the equator, so you can forget about the angular momentum of things on the earth. Is the container circular, with no obstructions on the bottom? Is the drain in the center? Is the water in the container absolutely still? Like if you sprinkled some dust on it and came back 24 hours later it would not have moved? If so, when you start the drain, the water will flow straight into the drain, with no vortex.
Anything that gives it the slightest angular momentum, in other words, any motion that is not toward or away from the drain, will be magnified as the water approaches the drain. It's the same as a spinning figure skater pulling in his or her arms and legs. Any weight that's pulled toward the center obeys Kepler's equal-area rule for orbits, so reduced radius results in increased angular velocity.
A: The velocity field of an incompressible fluid with no rotation obeys $\nabla \cdot v=0$ (incompressible) and $\nabla \times v=0$ (no rotational flow). As explained by Feynman, you can still have a fluid that moves in circles around a cylinder - it is just locally irrotational (zero vorticity; if you imagine putting in a tiny waterwheel in the fluid it will not turn). The velocity in this case scales as $v \propto 1/r$, which corresponds to a vortex. 
When the liquid drains, if it has some angular momentum then it must obviously start to circulate and the inward current will concentrate the vorticity. But even without angular momentum it is tricky to simultaneously maintain energy, momentum and mass conservation with a purely radial flow. So if the fluid starts circulating it gets an extra degree of freedom that allows it to satisfy all conditions, and it does not have to increase its vorticity to do so. Apparently the amount needed to make a swirling appear is fairly small. However, the swirl is temporary as angular momentum is gradually depleted by outflow. 
(This leaves out a fair bit of nonlinearity, see Andersen et al. for more stuff). 
A: Let me try to explain it. When water flows towards the hole, it chooses the shortest path (and we'll have next to no error if we assume that shortest path is a straight line). However, the Earth is rotating, and if you observe the straight-line motion of object from the rotating frame of reference, a motion relative to you will look curved. So, in fact the water is not rotating, it's the Earth "under" it that's rotating.
Coriolis force you mentioned does not exist. It's just the effect of non-inertiality of FOR.
For example, when the uniformly moving bus brakes, you feel like you are dragged forward, but in reality, you are just continuing the uniform motion (or, at least, trying to continue it), while the bus is moving with acceleration (deceleration in that case), making it's FOR non-inertial. And you feel that non-inertiality as fictional "force" dragging you forward (relative to the bus).
A: If the slightest asymmetry of the tub, air movement on top, convection current from uneven temperature etc., or disturbance causes a slight sideways movement of the water any where around the hole, it will deflect the incoming water slightly, as in the diagram. That will cause the incoming water to push the water around causing it to rotate. This will deflect the incoming water more, increasing the rotation, & so on.
It's like standing on a wheel. If you are right in the middle, where your weight is straight down towards the shaft, it will not rotate. But move slightly to the side, & it will start to turn, moving your feet to the side, which will make it accelerate more, moving your feet more to the side & so on. If the the tub was the size of a large lake and the drain hole big enough, and there was no wind to influence the movement of the water, the Coriolis effect MIGHT start it if no other disturbance starts it first.

