Q:

What common chemical/physical reactions would, in an atmosphere composed of ~20% Oxygen and ~80% Argon, behave differently than on Earth?

Ignoring how this sort of atmosphere would come about, what are some things that happen regularly on Earth that would be altered by this sort of atmosphere?
Example: Would firearms still ignite normally with that much argon present? Would humans have trouble breathing? How easily would light pass through without reflection/refraction?

A:

You've basically replaced nitrogen in Earth's atmosphere with argon, but left the ratios otherwise intact.
Argon is slightly denser than nitrogen; as a result, there would be slightly more light refraction in your argon atmosphere, but unfortunately I can't tell you what that would end up looking like to an observer. The speed of sound would also be ~10% slower, which isn't likely to be noticeable until timed.
Combustion (including firearms) would not be significantly affected, since that's really just oxidation of a fuel and you've left the ratio of oxygen in the atmosphere untouched. One thing that does change, however, is that you've eliminated less common byproducts of combustion, such as NO and NO2
The biggest problem is that you've eliminated the nitrogen cycle, which makes impossible the biology we know here on Earth. Nitrogen is crucial to every living organism we know, with atmospheric nitrogen being a staple of the process: Plants take it in, animals eat the plants, other animals eat those animals, and everything releases it back into the air during decomposition. Nitrogen is a critical piece of our biology, being a foundation of amino acids, and without it in the atmosphere we have no nitrogen cycle, and thus no (or at least insufficient) nitrogen to live on.
Unfortunately, argon is a noble gas, so it can't plausibly replace nitrogen in a hypothetical "argon cycle"-based biosphere. You have to either come up with a biology that doesn't depend upon nitrogen, or find a way to get it back in somehow (which, since it's a gas at anything above 77K, is pretty much impossible without it getting back into your atmosphere).
The good news is that argon is inert, so since there's adequate oxygen it's almost certain that humans could visit this planet and breath without needing full spacesuits and carrying heavy oxygen tanks. The slightly denser atmosphere would result in their voices sounding slightly deeper (think the opposite effect of inhaling helium, but much less pronounced), but otherwise they'd not really notice anything. They may just need to have access to foods that were part of the nitrogen cycle, and/or ship in nitrogen supplements for long-term habitation.

A:

I think plants would get into trouble. Plants need nitrogen compounds, which many of them get from bacteria that take the nitrogen from the air. Without nitrogen in the air, those bacteria cannot create nitrogen compounds, and thus those plants are in trouble.
Note that the nitrogen compounds in dung indirectly come from plants, too. So that's no way to get new nitrogen into the cycle. Ultimately all that nitrogen comes from the air.
And the nitrogen in artificial fertilizer is taken from the air as well, so unless there's another source available, even that won't work.
Note that Argon, as a noble gas, cannot replace nitrogen in chemical compounds.

A:

Argon is an inert gas. It takes a great deal of effort to get it to react with anything. 
Would firearms still ignite normally with that much argon present? 
Firearms (specifically, bullets) include their own oxidiser. That's why many guns will happily fire while underwater.
Would humans have trouble breathing? 
A little bit. Argon is ~43% more dense than Oxygen. We'd have a harder time breathing, but our bodies would very likely adapt to that change just fine. 
How easily would light pass through without reflection / refraction.
The refractive indices of Argon and Nitrogen are pretty similar. Interestingly, Argon's absorption spectra is quite a bit more lively than Nitrogen's. Despite it being a colourless gas, I imagine an atmosphere full of it would tint the sky with a tiny amount of green.
Argon's thermal conductivity is a lot lower than Nitrogen. Blowing on your hot coffee won't be as effective. 

