Pneumatic tires for bicycles have until recent years used inner tubes -- airtight, doughnut-shaped balloons. Motor vehicles used inner tubes until the 1960s, more or less, but over a period of a few years, tubeless tires became near-universal on motor vehicles.

Instead of having an inner tube, a tubeless tire seals directly to the flanges of the rim.

The rim must be airtight. The tire has a smooth seating surface that is pressed outward against the rim flanges by the air pressure inside. The process of mounting the tire requires soapy water or a special, soapy liquid, both to lubricate the tire to get it on the rim, and then to provide both a semi-sealing effect and a slippery surface for the tire's beads to mount on the steps of the rim.

Motor-vehicle rims have large flanges and so it is easy to establish a seal. The much smaller flanges of a bicycle rim make that more difficult. Also, bicycle rims have traditionally had spoke holes that extend through to the well (deepest part of the space between the rim's flanges). Rim tape protects an inner tube from sinking into spoke holes, but has not traditionally been airtight.

Since 2010 more or less, tubeless tires have come into use for bicycles. They were introduced initially as an aftermarket item, though they are appearing now on new high-end bicycles.

A Tubeless-Ready system can use a double-wall rim with normal spokes and recessed spoke holes, though not all such rims are tubeless-compatible. A special air-tight rim strip is used. This system is reported as being reliable on road bicycles with tires designed for a tubeless system, but there have been problems with other tires' rolling off the rim. A tubeless-ready system must use a slimy sealant both to make the initial seal and to seal against small punctures.

A True Tubeless system has a double-wall rim without spoke holes in the inner wall (well of the rim). Special, threaded spoke holes in the exposed face of the rim (closer to the hub) allow installation of threaded eyelets attached to special spokes, or the wheel may be of one-piece construction (usually, carbon fiber). A special wrench must be used to turn the eyelets. As of this writing, this system is sold only as a complete wheelset. A True Tubeless system may be run without sealant, though sealant improves its reliability.

The purported advantages of tubeless tires are  

• Without an inner tube, only an impact strong enough to punch a hole in the tire's fabric will cause a pinch flat. This is of special importance to off-road riders who run tires at low pressure.
• Sealant, if used, can plug small punctures without the need for patching -- desirable where flats are a major issue (Southwest USA with goat-head thorns, etc.). People who use tubeless tires report that they go for years without a flat. (But sealant can also be used with an inner tube, and then the sealant mess is almost entirely contained in the tube.)
• There is a claimed slight reduction in rolling resistance because of reduced sidewall thickness without an inner tube -- but on the other hand, the sidewalls of tubeless tires tend to be stiff, and stiff sidewalls increase rolling resistance, a known issue with road tires. There can also be rolling resistance due to sealant sloshing around. Jan Heine at René Herse Cycles has run tests and doesn't find any rolling-resistance advantage to tubeless tires. With fat, knobby off-road tires, rolling resistance is high in any case.
• There is a slight weight advantage, except that you have to carry a spare tube anyway in case of a puncture which won't seal.
• Except when there are problems with leakage, tubeless tires hold air longer than tires with inner tubes.
• Not all conventional tires stay on the rim if they flat or blow out. A tubeless tire generally will. That is a safety advantage, and allows a cyclist to limp home or to a bike shop -- though the tire will then probably have to be replaced.
• There's the prestige of buying into the latest and greatest technology;
• There's the fun (?) of fussing with the latest and greatest technology.

Disadvantages are:

• Tubeless tires are expensive -- a high-end product.
• The best tubeless-tire systems require buying new wheels, adding to the cost. Choices are limited. Retrofitting to rims not designed for tubeless tires is possible in some cases, but reliability problems have been reported.
• Road tubeless tires can't safely be run at as high pressure as ones with tubes, due to the risk of lifting off the rim (YMMV of course and tubeless for road bicycles is not yet a mature technology.)
• Tires with tubes win on ease of installation and replacement. The fit between a tubeless tire and the rim is tight. Special care must be taken to avoid tools' damaging rim flanges and tire beads; metal tire levers must never be used. Except for a skilled home mechanic, replacing a tire requires service at the bike shop. The trade-off is between more flat tires with easier replacement, and fewer with more difficult replacement.
• A blast of air from a compressor is often needed to seat the tire on the rim. A frame pump works too slowly to pop the tire into place to form a seal. Some tubeless-tire manufacturers claim that a floor pump will do -- particularly with a skinny tire which inflates quickly. Some newer frame pumps are designed to release a blast of air, though. A CO₂ cartridge might work, especially with a skinny tire, but the tire must be deflated and reinflated during installation, and that takes two cartridges. The valve core must be removed during the first inflation to allow rapid airflow.
• Because special sealing liquid must be sprayed or painted on the tire beads to form the initial seal, you would have to carry this with you to re-install a tubeless tire on the road. If the hole in the tire is too large for sealant to plug, then sealant must be cleaned off, and patching is iffy. In the event that the tire has a gash which requires a boot, there must be a tube inside to hold the boot in place. (With an inner tube, booting the tire and patching or replacing the tube are separate issues.) All in all, if you get a flat on a ride, and sealant doesn't stop the leak, you had better carry a tube -- and tools to remove the tubeless valve, which is attached to the rim -- or have a team car with replacement wheels following you. Lifting the tire to install the tube is difficult and messy.
• Sealant, needed to prevent air leakage with some tubeless tire/rim combinations, dries out and must be replenished every few months.
• There have been claims of the ability to run a tubeless tire at lower pressure. This is in our opinion largely marketing spin. One manufacturer claims that its road tires can be run at 13% lower pressure. That is only marginally significant. Using a somewhat fatter tire allows lower pressure, increases ride comfort, and has been shown not to increase rolling resistance. A fatter tire with a tube, properly inflated, also will avoid pinch flats, and by keeping the rim higher off the road, also avoid rim damage.
• Is running at lower pressure an advantage, or is it a limitation? There are reported cases of tubeless tires' blowing off rims if inflated hard. Neither do they hold air as well at low pressure (for example, with off-road tires). Slight movements of the tire against the rim can cause a tire to "burp". This occurs when the tire casing is subjected to greater lateral load than it will stand, so that a short section of the sealing surface opens up and lets out air. The cyclist may detect this only later by the presence of some sealant along the rim/tire joint. However, once a tubeless tire's pressure has decreased so it no longer forms a seal, it will suddenly deflate. A friend of mine broke his hip during a cyclocross race that way.
• Any loose item inside the tire will roll around and possibly make noise. With an inner tube, air pressure holds debris in place.

It has been over 50 years since tubeless tires came to dominate the automotive market, and they are only now being introduced for bicycles. That isn't a question of lagging technology as much as it is one of what is most practical.

Top-quality bicycle equipment has become far more expensive over the past few decades, even after adjusting for price inflation. The authors of this article look at today's prices for top stuff and think, "It isn't worth it."  But the bike companies have customers who don't think the way we do.  People enjoy spending ever-more money on their hobbies, pursuing ever-diminishing returns. From wine snobs to folks buying stuff to make their Ford Mustangs go faster, that's how some people behave.

Even if people don't know why, or whether, something is better, it gives them something to talk about, and about which to feel pride of ownership.

The authors of this article developed our tastes in an era when it was possible to maintain almost anything on a bike with a few simple tools.  We took pride in being able to perform minor repairs and complete our day's ride. Today, a torque wrench and the expertise to use it properly are mandatory with the fancier equipment, making a cell phone the preferred tool in case of a problem when out on a ride.

All in all, tires with inner tubes are by far the best choice for bicyclists interested in self-sufficiency and reasonable value for money. This might change with improvements in technology, but that is our judgment now.

So if we are puzzled by the "appeal" of tubeless tires, it's because we are on a different operating system from the intended customers.