last updated 20 Mar 2022
Subwoofers use speaker drivers (woofers) typically between 8-inch and 21-inch in diameter. Subwoofers typically play 20Hz to 200Hz
Why do you need a subwoofer.
Bookshelf speakers typically have a frequency response range that is approximately 50 or 60 Hz to 20 kHz. As you can see below there are at least 7 musical instruments with a lower frequency range.

See Subwoofer - Wikipedia

Most subwoofers have a frequency response range of 30Hz to 200Hz
For normal young ears the entire audible range of sounds extends from about 20 to 20,000 hertz
Music can go as low as 18 Hz (which is about the pitch of the lowest rumbling notes on a huge pipe organ with 32-foot bass pipes)

Musical Instruments with low bass at  Sound in tech.
Piano A0 (28 Hz) to C8 (4,186 Hz) Bass C1 (33 Hz) to C5 (523 Hz) Picolo C5 (523 Hz) to B7 (3,951 Hz) Guitar (Bass) 4 string E1 (41 Hz) to C4 (262 Hz) Barritone Sax C2 (65 Hz) to A4 (440 Hz) Harp (28 Hz) to (32,000 Hz) Tuba (45 Hz) to (380 Hz) Large Pipe Organ with a 32 foot stop in the pedal (16 Hz)
Frequencies within the range of 4 to 16 Hz are felt but not heard.

See: 10" vs 12" Subwoofers (The Real Difference) | BoomSpeaker
Size is the speaker element not the box.
10" subs sound punchy while 12" subs sound boomy.
A 10" sub has less distortion
A 12" sub is louder and better for a large room.
Examples: (Manufacturer specs - List response but not SPL(dB)

10" Polk Audio PSW10         $159 40-160 HZ 
10" Elac SUB1010             $130 37-200 Hz
10" JL Audio 10W3v3-4        $380 25-250 Hz
10" Klipsch Reference R-10SW $200 32-120
12" Polk Monitor XT12        $350 24-160 Hz
12" ELAC Debut 2.0 SUB3030   $650 25-150 Hz
12" Jamo C912                $205 29-125 Hz
15" Kicker 45L7R152          $429 20-100 Hz
  (element only no enclosure)  
15"  Dayton Audio RSS390HF-4 $270 18 to 800Hz
   (element only no enclosure 
Are frequency responses specifications consistent between manufacturers:
See Frequency Response Specs

Higher end subwoofers have their own powered amplifiers.
'Passive subwoofers' have a subwoofer driver and enclosure, but they do not include an amplifier. Crosover:
Crossovers limit the frequency response of a speaker. Subwoofers typically employ a crossover called a "low-pass filter." A low-pass filter only permits source frequencies of a certain hertz and below to reach the subwoofer. Professional concert sound system subwoofers typically operate below 100 Hz,
THX-certified systems operate below 80 Hz.

DAC - Digital to Analog converter
Hz - Cycles per second
SPL - sound pressure level, and it's measured in decibels (dB)
THD - Total harmonic distortion
THX - Tomlinson Holman's eXperiment. Holman created THX while working with Lucasfilm studio to devise a new standard for audio reproduction.

Erin's Audio Corner had the most comprensive testing I could find.

Erin recommends multiple subwoofers to provide a smoother in-room response.
YouTube presentations:
Budget <$200 10 inch subwoofers | Erin's Audio Corner
10" down to 
A 10" subwoofer is good with bookshelf 10" speakers
1. Elac SUB1010 $129 powered 60 w  box 12" w 14" 
2. Polk sw10 $150

Budget ($175-$300) 12 inch subwoofers | Erin's Audio Corner
Down to  30Hz
1. Infinity R12 $175 (not available 2022)
2. Jamo C912 $250
3. Eclipse 12W

Pay $400-$600 to go lower than 20 Hz

See Frequency Response Specs
Audio - stereo - Home Theater (Surround Sound) in home and garden.
Sound in tech.