The Earth rotates to the east at an effectively constant angular velocity, but different latitudes have different linear speeds. A point at the equator has to go farther in a day than a point in Ohio, so it must go faster.
However, when an object starts to move north or south and is not firmly connected to the ground (air, artillery fire, etc) then it maintains its initial eastward speed as it moves. An object leaving the equator will retain the eastward speed of other objects at the equator, but if it travels far enough it will no longer be going east at the same speed the ground beneath it is.
The result is that an object travelling away from the equator will be heading east faster than the ground and will seem to be forced east by some mysterious force. Objects travelling towards the equator will be going more slowly than the ground beneath them and will seem to be forced west. In reality there is no actual force involved, the ground is simply moving at a different speed than the object is "used to".
Consider the diagram to the right. The orange arrow represents some object sent north from the equator. By the time it reaches the labeled northern latitude, it's gone farther east than a point on the ground would have, since it kept its eastward speed from where it started. Similarly, the yellow arrow started away from the equator at a slower eastward speed, and doesn't go as far east as the ground at the equator...seeming to deflect west from the point of view of the ground.
Again begin with an object at a particular latitude where the Earth moves east at a certain speed. Now move it east or west so that it has a speed different from that of the Earth below.
(Quick pause to explain centripetal acceleration in a nutshell.)
(Centripetal acceleration is defined as the acceleration needed to keep an object moving in a circle at a particular radius. In the case of objects on Earth, the radius is a line perpendicular to the axis of Earth's rotation, and the acceleration is provided by the component of gravity in that direction.)
(The basic effect of all of this, however, is that if you go faster than "allowed" by your centripetal acceleration you start to fly outward and if you go slower you drop inward. A good demonstration of this is to spin a ball attached to an elastic cord: spin faster and it goes outward, spin slower and it comes in.)
As a result, objects moving east want to fly outward into space and objects moving west want to drop towards the axis of the Earth. However, gravity keeps most objects from flying away and the ground stops things from falling straight to the axis. Most objects will be confined to a few mile thick layer at the surface, and while they can rise and fall some, the best way to change their radius is to deflect north or south.
Eastward-bound objects will try to go straight out, but as shown in the diagram to the left, they will head for the equator as the best way to increase their radius out from the axis. Westward-bound objects will "skid" off the ground and head away from the equator to where the radius is smaller. In cases where the motion isn't enough to make the object run into the upper or lower limits, simply moving away from the axis will make the position above ground move farther south, and moving towards the axis will take an object farther north.
The general result of any one of these deflections is that something in the Northern Hemisphere moving along in one direction will be deflected to its own right with respect to an observer on the ground. In the case of a low pressure system where everything is moving towards the low, it creates a spinning vortex, as seen on the right.
Because it largely depends on how large the difference is between an object's velocity is and the ground's velocity, the effect is really only significant at high speeds (either type) or long distances (north/south especially). The angular velocity of Earth is 360 degrees per day, or .2 microradians per second, quite small. Even at fairly high wind speeds found in typhoons (40 meters per second) the Coriolis Effect generates a deflection of only about ten microns per second squared. Over an hour, this is a total deflection of about 100 meters...over a day a deflection of almost 40 kilometers. It adds up, but it takes time.
How slow is slow enough? Well, a quick order of magnitude calculation can be used here. Figure that after the first second of draining a sink has gotten set in whichever way it's going to spiral down the sink...it only gets stronger after it starts moving (since the velocity towards the drain picks up). Figure an average particle is at a radius of ten centimeters from the drain, so a micron per second corresponds to about 2 microradians per second angular velocity that can be changed before things get out of hand. Or about 100000 seconds per rotation....one rotation per day, roughly. Give or take an order of magnitude and one full rotation per hour is all the Coriolis Effect can reverse. Even take two orders of magnitude and you still can have water spinning at once every few minutes and still be spinning the "wrong" direction enough to ignore the Coriolis Effect and go down the sink its own way. This is certainly "not visibly moving."
Of course, many sinks will exhibit no visible spin at all if they are small enough that no relic spin can work up to a good whirl and will only spiral if the water is let out while still rotating very visibly.
As to whether there's a conspiracy among sink manufacturers to sell only clockwise-encouraging faucets south of the Equator and counterclockwise faucets north of it is a topic for another day. The reader is encouraged to search the journal Science where I am told a pair of serious studies are presented (unfortunately, I don't know which issues the articles are in, or even which year...my information is still secondhand). The results were that after you carefully control all the variables (use a large wooden tub, control the temperature throughout the tub, have the drain be a tube extending up into the tub to avoid friction effects with the tub walls, start the water off spinning the "wrong" way, etc) and wait 18 hours for the water to settle down, water does indeed spiral down the sink opposite directions in the two hemispheres. But this effect is so subtle that it wouldn't ever be seen in your bathroom sink.