What is one common use for PIN diodes?

RF switch.

From dallasepperson:

Under zero or reverse bias, a PIN diode has a low capacitance. The low capacitance will not pass much of an RF signal.

Under a forward bias of 1 mA, a typical PIN diode will have an RF resistance of about 1 ohm, making it a good RF conductor. Consequently, the PIN diode makes a good RF switch.

From Wikipedia's article Diode

"A diode is a two-terminal electronic component that conducts current primarily in one direction (asymmetric conductance); it has low (ideally zero) resistance in one direction, and high (ideally infinite) resistance in the other."

From Wikipedia's article PIN diode

" PIN diode is a diode with a wide, undoped intrinsic semiconductor region between a p-type semiconductor and an n-type semiconductor region. The p-type and n-type regions are typically heavily doped because they are used for ohmic contacts.

The wide intrinsic region makes the PIN diode an inferior rectifier (one typical function of a diode), but it makes it suitable for attenuators, fast switches, photodetectors, and high-voltage power electronics applications."

For more information and application to radio transreceivers, please see the Industrial Electronics site, for the article on
PIN diodes and their uses [RF Circuit Electronic Design]

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What is a common use of a hot-carrier diode?

VHF and UHF mixers and detectors.

From jriechel:

A hot carrier diode is also known as a Schottky Barrier Diode. It's a fast switching diode (due to low capacitance) with a low forward voltage drop.

"The fast switching function is effective as a mixer or detector at VHF/UHF frequencies." -- david_abraham

Note also that Schottky Barrier Diodes are also commonly used in power supply circuits because the lower forward voltage drop reduces power dissipation and improves efficiency. However, they don't support high current loads.

For more information, please see Wikipedia's article on Schottky diode

Also, see the Elprocus site for the article Schottky Diode Working and Applications

And, view the YouTube video from the Afrotechmods channel called What is a schottky diode?at 6:01 minutes.

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Structurally, what are the two main categories of semiconductor diodes?

Junction and point contact.

For more information, please see Electronics Tutorials site for the article PN Junction Theory.

An excellent description of diodes classification is at Rohm site in the article
Outline of a Diode ＜Classification＞

Also, a great overview of diodes is at Electroniclinic site, in the article
Semiconductor Diodes and its Types

And, well-illustrated information on Electronics Hub under the article Types of Diodes | Small Signal, LED, Schottky, Zener

See Resources PCB Cadence article on Forward Bias vs. Reverse Bias and their Effects on Diode Functionality

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What special type of diode is capable of both amplification and oscillation?

Tunnel diodes.

From Wikipedia's article on Tunnel diode

"A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively "negative resistance" due to the quantum mechanical effect called tunneling."

and

"Tunnel diodes have a heavily doped positive-to-negative (P-N) junction that is about 10 nm (100 Å) wide. The heavy doping results in a broken band gap, where conduction band electron states on the N-side are more or less aligned with valence band hole states on the P-side. "

For more information, please see Electronics Notes site for the article Tunnel Diode: theory & characteristics

Also, please see well-illustrated article on Electronics Hub site Tunnel Diode – Working, Characteristics, Applications

And, WatElectronics site for the article What is Tunnel Diode : Working & Its Applications

Please see Electrical 4 U site for the article Tunnel Diode: Definition, Characteristics & Applications

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What type of semiconductor diode varies its internal capacitance as the voltage applied to its terminals varies?

Varactor diode.

From kv0a:

In electronics, the following:

• a varicap diode
• varactor diode
• variable capacitance diode
• variable reactance diode
• tuning diode

are types of diode whose capacitance varies as a function of the voltage applied across its terminals.

Please see Wikipedia's article on Varicap

Hint: "V" in voltage, "V" in varactor diode.

Also think variable capacitor diode.

For additional information, please see Electronics Notes for the article on
Varactor Diode or Varicap Diode

Also see Electrical 4 U for the article on Varactor Diode

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What is the principal characteristic of a tunnel diode?

Negative resistance region.

From Wikipedia's article on Tunnel diode

"A tunnel diode or Esaki diode is a type of semiconductor diode that has effectively "negative resistance" due to the quantum mechanical effect called tunneling."

and

"Tunnel diodes have a heavily doped positive-to-negative (P-N) junction that is about 10 nm (100 Å) wide. The heavy doping results in a broken band gap, where conduction band electron states on the N-side are more or less aligned with valence band hole states on the P-side. "

For more information, please see Electronics Notes site for the article Tunnel Diode: theory & characteristics

Also, please see well-illustrated article on Electronics Hub site Tunnel Diode – Working, Characteristics, Applications

And, WatElectronics site for the article What is Tunnel Diode : Working & Its Applications

Please see Electrical 4 U site for the article Tunnel Diode: Definition, Characteristics & Applications

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