How does this vortex form inside a jet engine?
Whenever there is under-pressure in air, the air flows along the gradient of this under-pressure, i.e. into the direction of the source of the under-pressure. (In first order.)

If there's a wall (or the ground) in the way, air will flow along the surface of this obstacle and will change its direction at some point, as shown in this sketch:



Now, imagine the whole air around the engine is rotating slowly for some reason around the symmetry axis in the sketch.  

From physics, you may know the conservation of momentum, or in this case, of angular momentum. It says that the product of distance to rotation axis times RPM of this rotation is constant for a certain particle. If a particle reduces its distance to the axis, its RPM increases.
This is like the spinning ice skater, who rotates faster when he pulls his arms to his body (to the axis).

This law also applies to packets of air, which are sucked to the axis, though internal friction brakes the air to some extent.

The fast rotating air near the axis encounters a centrifugal force, i.e. the air is pulled away from the axis, forming a region of strong under-pressure. In this region, the air expands and so, cools down. This can cause condensation of water, if the humidity of the air is high enough. And that's what you can see on your pictures.

There's still one question unanswered: I wrote, this happens if the surrounding air is rotating slowly. This usually happens, because the wind is deflected from the aircraft itself, which means it changes its direction, which is some kind of rotation.

And of course, a strong, non-uniform air flow from the rear, as generated by the reverse thrusters, can cause more / stronger rotating air, which results into stronger vertices. So, it's no wonder when they appear (or better: become more visible) in this case.

Here is a nice picture of a simulation, which also shows that the fuselage can be the wall where the vortex is created.


Source: http://gallery.ensight.com

It seems the wind comes from the camera position, or a bit more right of it. It gets deflected by the fuselage and flows along it, which explains the straight lines coming from the rear and the curved ones coming from the front left of the aircraft. If you imagine the air flow along / under the fuselage, you may also understand why the two vertices are rotating in different orientations.

As the strong under-pressure inside the vortex, it may also suck in dust or even larger objects, which can damage the engine. For aircraft landing on unpaved airfields, unpaved strip kits may be available, which can contain a vortex dissipator:


Source: http://www.b737.org.uk

This is a nozzle blowing bleed air into the region where vortices could apper, reducing the under-pressure and so the risk of ingesting objects from the ground.



Notes:

You can see something similar in the sink of your bathtub. As soon as there is a little rotation, it is amplified above the sink, the centrifugal force pushes the water aside, and forms this fast rotating vortex.

A tornado is a giant version of an inlet vortex. The air is moving along ground over far distances, until it moves upwards at some point, and a slight rotation leads to this fast rotating hose sticking to the ground and sucking in everything due to the strong under-pressure inside.

And a bit off-topic:

Sometimes, you can see vortices at the wing tips or at the edges of engaged flaps of aircraft flying through humid air. The reason is similar: The air is put into rotation due to the higher pressure below and the lower above the wings. (But that's not exactly the same as inlet vortices.) And sometimes, you also see fog forming above the wing just because of the under-pressure.
On this picture, you see both, and may imagine the rotation for the second vortex from the right.


Source: http://avioners.net