Why do some radio antennas require a path to ground and some do not?
Apologies for the long answer - its a complicated subject. I've tried to focus on the main points. Its not meant to be a comprehensive treatment of aerials.

Ground and Earth.

If by 'ground' you mean physically connected to the Earth then the answer is plainly no, otherwise communications satellites or space probes wouldn't work. If by 'ground' you mean a point in the circuit that all other voltages in a circuit are referenced to (0V), then the answer is yes.

Aerials work by transmitting or receiving an electromagnetic wave. (See Maxwell, Hertz, et. al)

The EM wave - aerial types and shapes:

EM Waves can be detected from their electric field or their magnetic field  (or both). The wave can also be polarized so aerials can be optimized for this polarization as well. The 'gain' of an aerial can be increased by making it directional, adding extra elements to it or using shaped reflectors. Hence the wide variety of shapes and sizes and plain wacky looking designs to optimize reception (or transmission) at different wavelengths. 



The 'simple' monopole.

A vertical aerial (the one commonly used with a crystal set) detects the electric field. This moves charge 'up and down' the aerial producing a small but measurable alternating current. The effect is greatest when the physical dimensions of the aerial match the wavelength of the EM wave. (see resonance effects) It does not require that the aerial is connected to anything. 

This type of aerial is omni-directional.

Coupling the (weak) signal to the receiver requires tuning and impedance matching. For a aerial this is performed with the LC tuned circuit (Tank). The tuned circuit will magnify one selected narrow frequency band ( see Q factor). The tuning coil can be tapped (as in an auto transformer) or a separate coil can be used to optimize the loading on the aerial as some of the energy is transferred to the 'radio' circuit. Taking too much energy would damp the response. Keeping a high Q factor allows you to easily separate individual transmissions. (Selectivity)

To maximize this effect the length of the aerial should be related to the wavelength of the signal. The ground (earth) acts as a reflective surface (not as a return conductor) and can double the effective length of the aerial, producing a dipole aerial. This ground plane effect can be improved by using a sheet of metal (good conductor) or even wire.



The common yagi array (TV aerial) demonstrates this idea. The signal is taken from a folded dipole. The other elements are dipoles (horizontally or vertically polarized or both). These act as passive aerials and retransmit the incoming wave with a phase shift so that the add to the signal strength at the main receiving aerial. The yagi is sensitive to direction.



Portable Radios.

The aerial pick up in a a portable radio is a coil wrapped around a ferrite bar. ( a loop aerial) and detects the magnetic field component of the EM wave. This changing magnetic field induces a voltage in the coil. The coil forms a tuned circuit with a variable capacitor. A second 'output' coil or perhaps a 'tap' on the primary coil impedance matches the amplifier circuit without loading the tuned circuit, producing a sharply tuned aerial.

Loop aerials are very directional as you will easily observe by rotating the portable radio through 360 degrees. 



Note that one end of the tuned circuit is connected to the 'ground' or 0V of the rest of the radio circuit. You could also add a second aerial (a length of wire or 'whip' aerial) usually connected by a small capacitor to the 'hot' end of the coil or perhaps a separate coil on the ferrite bar. This pulls in the electric field component of the signal and improves reception.