Airplane contrails are real, but we know what they are: water vapor and small amounts of other combustion products created by jet engines. The other pollutants may be somewhat harmful, but they are basically the same thing as produced by any other form of combustion. So what are chemtrails? Wikipedia sums it up nicely:
The chemtrail conspiracy theory is the unproven belief that long-lasting trails, so-called "chemtrails", are left in the sky by high-flying aircraft and that they consist of chemical or biological agents deliberately sprayed for sinister purposes undisclosed to the general public. 
When I say chemtrails may be real in the future, I'm really only making the claim that high-flying aircraft will be deliberately spraying chemicals that are not simply a normal by-product of jet engine operation. However, it won't carried out for "sinister" reasons and it won't be done in secret. Instead it will only be done after a great deal of public discussion and scientific deliberation.
The chemical in question? Sulfur dioxide.
In 1991 Mount Pinatubo in the Philippines produced the second largest volcanic eruption of the 20th century. Besides massive amounts of volcanic ash, Pinatubo injected an estimated 17 million metric tons of sulfur dioxide into the atmosphere. This caused measurable global cooling of about four-tenths of a degree Celsius from 1991 to 1993. The effects of the eruption lasted for about three years. 
The Mount Pinatubo eruption and its worldwide effect on the climate has raised the possibility that we could intentionally inject sulfur dioxide into the atmosphere to counter-act the effects of global warming. This kind of large scale intentional intervention is frequently called "geoengineering" although Wikipedia prefers the term "climate engineering". 
Seventeen million tons of sulfur dioxide might seem like a lot, but over one one-year period in 2013 and 2014 U.S. airlines alone consumed about 50 million tons of aviation fuel. If jet aircraft were burning fuel (at altitude) as sulfur-heavy as marine bunker fuel (3 or 4%) then we could be talking about the equivalent of a Mount Pinatubo eruption every couple of years. 
A massive climate engineering project to inject sulfur dioxide into the stratosphere is clearly plausible, but is it a good idea? We can do it, but should we? There are two big questions we need to answer. How useful is it? And: How risky is it?
Earth's surface has warmed by about 0.85 degrees Celsius from 1880 to 2012. In comparison, the effects of the 1991 Mount Pinatubo eruption lasted for a couple of years and caused global cooling that was estimated to be about 0.4 degrees Celsius. Pinatubo-scale sulfur dioxide injection -- if sustained -- could offset a large proportion of the global warming that we've already experienced. Of course sulfur dioxide injection only treats the symptoms of global warming, not the root cause. But it sure looks like it could provide some real value as an intermediate stop-gap solution while other more permanent solutions are brought online. 
The data from the Mount Pinatubo eruption tells us that sulfur dioxide in the stratosphere can cause cooling comparable to the global warming we've seen so far. But it tells us something else as well: The effects of stratospheric sulfur dioxide have a limited lifetime, on the order of just a couple of years. This is bad in the sense that to be useful, the injection program has to be sustained. But it is good -- very, very good -- in the sense that it reduces risk. The program could be scaled up, the impact could be measured, and, if the side-effects are too serious, it can be scaled back down again very rapidly. This greatly reduces the risk of the undertaking.
Global warming melts polar ice packs and ice sheets, exposing darker water and land underneath. The reduction in reflectivity results in more of the sun's energy being absorbed by the earth, resulting in warmer temperatures, resulting in more ice melting, resulting in more land and water being exposed, resulting in even warmer temperatures. This is often incorrectly referred to as a "negative feedback loop". In reality it is a positive feedback loop with negative consequences. There are other feedback loops of great concern. For example, as arctic regions get warmer, methane that has been trapped in permafrost and undersea clathrates is released. And since methane is a powerful greenhouse gas, it too can cause a positive feedback loop. 
The offset cooling provided by stratospheric sulfur dioxide injection would interfere with these feedback loops. This could be particularly valuable in the case of methane, which is a more powerful greenhouse gas than carbon dioxide but which also has a quite short lifetime in the atmosphere.
Stratospheric sulfur dioxide injection is not a panacea. But what it can do is buy us time. It can buy us time to replace fossil fuels with cleaner sources of energy that don't release carbon dioxide into the atmosphere. It can buy us time to replace less energy efficient technologies with more efficient ones. It can buy us time to deploy large scale climate engineering projects that do remove carbon dioxide from the atmosphere. It can buy us time to deal with the immediate consequences of global warming and climate change (and we are simply going to have to deal with some of them), and indeed it can buy time for developing economies to, well, develop. Because everything we need to do is going to be easier if everybody has more money.
Now, to get back to my original point. I don't know for sure that we're going to be using jet aircraft to inject sulfur dioxide into the atmosphere, and I'm no expert anyway. But I think we're going to do it. And if we do, the result will be, quite literally, "chemtrails".
-  https://en.wikipedia.org/wiki/Chemtrail_conspiracy_theory
- The chemtrail conspiracy theory is the unproven belief that long-lasting trails, so-called "chemtrails", are left in the sky by high-flying aircraft and that they consist of chemical or biological agents deliberately sprayed for sinister purposes undisclosed to the general public.
-  https://en.wikipedia.org/wiki/Mount_Pinatubo
- The second-largest volcanic eruption of the 20th century, and by far the largest eruption to affect a densely populated area, occurred at Mount Pinatubo on June 15, 1991.
- The injection of aerosols into the stratosphere is thought to have been the largest since the eruption of Krakatoa in 1883, with a total mass of SO2 of about 17,000,000 t (19,000,000 short tons) being injected—the largest volume ever recorded by modern instruments.
- This very large stratospheric injection resulted in a reduction in the normal amount of sunlight reaching the Earth's surface by roughly 10% (see figure). This led to a decrease in northern hemisphere average temperatures of 0.5–0.6 °C (0.9–1.1 °F) and a global fall of about 0.4 °C (0.7 °F).
- The stratospheric cloud from the eruption persisted in the atmosphere for three years after the eruption.
-  https://en.wikipedia.org/wiki/Climate_engineering
- Some proposed climate engineering methods employ methods that have analogues in natural phenomena such as stratospheric sulfur aerosols and cloud condensation nuclei. As such, studies about the efficacy of these methods can draw on information already available from other research, such as that following the 1991 eruption of Mount Pinatubo.
- Climate engineering, commonly referred to as geoengineering, also known as climate intervention, is the deliberate and large-scale intervention in the Earth’s climatic system with the aim of limiting adverse climate change.
-  https://en.wikipedia.org/wiki/Environmental_impact_of_aviation
- U.S. airlines alone burned about 16.2 billion gallons of fuel during the twelve months between October 2013 and September 2014.
- According to https://en.wikipedia.org/wiki/Jet_fuel, jet fuel ranges from 775-840 g/L.
-  https://en.wikipedia.org/wiki/Fuel_oil
- Governing bodies (i.e., California, European Union) around the world have established Emission Control Areas (ECA) which limit the maximum sulfur of fuels burned in their ports to limit pollution, reducing the percentage of sulfur and other particulates from 4.5% m/m to as little as .10% as of 2015 inside an ECA. As of 2013 3.5% continued to be permitted outside an ECA, but the International Maritime Organization has planned to lower the sulfur content requirement outside the ECA's to 0,5% m/m.
-  Math
- How much sulfur dioxide could be produced by burning 16.2 billion gallons of fuel which is 3.5% sulfur by weight?
> 16.2 billion gallons
> 61.3 billion liters ; 3.78541 liters per gallon
> 47.5 billion kg ; 775 g/l or 0.775 kg/l, the lower figure for jet fuel
> 47.5 million tons ; 1000kg per metric ton
> 1.66 million tons ; tons sulfur, assuming 3.5% sulfur by weight
> 3.32 million tons ; sulfur dioxide is about 1/2 sulfur by mass (32.06 / 64.066)
- Answer: 3.32 million tons sulfur dioxide
- This number is at best an estimate. Aircraft would only burn sulfur heavy fuels at altitude, so not all fuel consumed would produce stratospheric sulfur dioxide. On the other hand, average jet fuel density is no doubt greater than 775 g/l, and sulfur concentrations of up to 4.5% have been reported for marine bunker fuel.
- Most importantly, U.S. air travel only accounts for a fraction of total air travel -- a large fraction, no doubt, but probably less than half.
> 3.32 million tons ; assume U.S. aircraft inject 3.32 million tons of sulfur dioxide
> 6.64 million tons ; assume European and Asian aircraft can inject a similar amount
> 13.3 million tons ; assume a 2 year time horizon
-  https://en.wikipedia.org/wiki/Global_warming
- The global average (land and ocean) surface temperature shows a warming of 0.85 [0.65 to 1.06] °C in the period 1880 to 2012, based on multiple independently produced datasets.
-  https://en.wikipedia.org/wiki/Ice-albedo_feedback
- Conversely, warming tends to decrease ice cover and hence the albedo, increasing the amount of solar energy absorbed, leading to more warming.
-  https://en.wikipedia.org/wiki/Arctic_methane_emissions
- Arctic methane release is the release of methane from seas and soils in permafrost regions of the Arctic, due to deglaciation. While a long-term natural process, it is exacerbated by global warming. This results in a positive feedback effect, as methane is itself a powerful greenhouse gas.