The ocean circulation is driven by the transfer of heat, water, and momentum across the air-sea interface. Heat transfer is accomplished through (1) latent heat flux, (2) sensible heat flux, and (3) radiation. Water transfer is accomplished through evaporation and precipitation. Momentum transfer takes place through the frictional effects of winds on the ocean surface.

If we imagine that ocean circulation is only responding simply to winds (ignoring that the ocean is in rotation), we would expect simple gyres to form in each ocean basins in response to the tropical easterly trades and midlatitude westerlies with the north-south branches along the coastlines to conserve mass. The gyre centers would be at about 30 N and 30 S in the center of the basin, and water would pile up where the eastward and westward moving branches approached land (see Fig 5-1 and the first row in the figure below) 

and it the surface would be low in the opposite corners. Now if we account for the influence of Earth's rotation we create a net transport of the surface water at a 90 deg angle to the wind, we see that the water piles up in the center of the gyre, instead of at the corners. However this still ignores conservation of vorticity. The middle row of pictures would be accurate if the Earth were a rotating disk and not a sphere (hence the coriolis force would be constant). But on a sphere, the coriolis force increases with latitude, and this causes the branch of the gyre that is near western boundaries to be concentrated in what is known as a western boundary current (lowest row). The explanation given in the text is decent. I will try to fill in a few gaps below. Before I go into that explanation, I want to describe geostrophic current and coastal and equatorial upwelling.

Like the geostrophic wind, the geostrophic current flows perpendicular to the two forces that balance to allow this steady flow. The forces are the pressure gradient force and the coriolis force (see figure at right). The geostrophic current would flow around the water that piles up in the middle of the gyre, except friction slows it down a little, which makes the actual current flow diverge outward a little.

The figure on the left shows how Ekman transport causes coastal upwelling. The "coriolis effect" in the figure on the left is our Ekman transport. In the upper panel, the Ekman transport pulls water away from the coast. This water is then replaced by upwelling deeper water, which is colder. Locations in the northern hemisphere with persistent northerly winds just offshore have cold water at the coast, like in our state and in Oregon and California. The picture immediate below shows how Ekman transport due to the easterly trade winds causes upwelling along the equator.

Vorticity is the tendency of something to rotate. It is defined so that the tendency to rotate clockwise is positive and counterclockwise is negative. If you stood at the north pole you would spin clockwise at the rate of once per day, and so the Earth imparts positive vorticity on you. If you stood at the south pole you would rotate counterclockwise and so the Earth imparts negative vorticity on you. This rotation brought about due to Earth's rotation is called planetary vorticity. I don't like this explanation of planetary vorticity because it is misleading in several ways. For one thing it does nothing to explain how Earth imparts "rotation" elsewhere without resorting to mathematics. Another way to think about it is to consider the path a fluid parcel takes when it is given a shove in the horizontal plane. Recall that the coriolis effect deflects parcels to the right in the NH. The path is actually curved, which can be considered a rotation. Ben pointed out that this argument gives the rotation in the wrong direction because planetary vorticity is positive in the NH where it is counterclockwise. Ben is absolutely right. I'm at a loss to give a physical explanation for this thing called rotation. Interestingly, planetary vorticity and the coriolis effect have magnitudes that scale the same way with latitude, even though a parcel need not be in motion to have a tendency to rotate, while the coriolis effect only applies to a moving parcel. 

Relative vorticity is easier to understand. It is the tendency of a fluid flow to cause rotation. The following figure shows how:

The black arrows show the sense of rotation that the flows create. The curved flow on the upper left causes a counterclockwise rotation, which is in the same sense as the planetary rotation in the NH, so it is called positive.

The sum of planetary and relative vorticity is conserved (or close to it). This conservation helps us understand why western boundary currents are focused near the far western side of a basin. The subpolar gyre has negative vorticity (like red arrows on the upper right). Along the western boundary, the flow is moving northward and so it gains positive planetary vorticity. In order to conserve the vorticity, the relative vorticity must become even more negative so the sum (relative plus planetary) does not change. This causes the flow to speed up, especially near the center of the gyre (think of a skater bringing his arm towards his body to rotate faster). The current becomes narrow to conserve flow. The center of the gyre and this narrow fast moving current moves closer to the western boundary to increase the friction and slow down a little.  The opposite occurs in the eastern boundary.