In the case of COVID19, the virus is carried in water droplets expelled by an infected person when they breathe, talk, etc.

A mask helps by blocking those droplets, both from exiting the infected person and from entering the people nearby. One way to stop small droplets is to make any holes even smaller. This means that one of the best "characteristics" is a tight weave.

High-thread-count cotton, a natural material, works well because it can be woven tightly and the fibers themselves have additional structure that can further help to block droplets. On the other hand, pure synthetic materials have relatively smooth fibers that do not block as well. Another way to stop the droplets is to attract them to something. This is essentially by static electricity. Synthetics and blends tend to be better than natural fibers for this.

The most effective masks have layers of multiple types of materials - one (or more) with tight weaves and thus small holes, and others to provide electrostatic attraction.

Of course, none of that matters if the masks are designed poorly or used improperly. The mask needs to fit snugly around the edges (e.g., minimal gaps around the nose and chin) to prevent inhalation of droplets with air that travels around the mask.

Sources: MIT , ACS , Boston Hospital .

To stop a virus or another harmful substance, it is necessary to use materials which:
(1) have holes that are too small to let the harmful substance through,
(2) are electrically charged so that harmful substances get stuck to the material rather than going through, and
(3) are waterproof so that the virus cannot seep through along with water.

Stopping a single, isolated virus particle is very difficult and requires specialized materials, like the ones used to make professional N95 or N99 masks. These materials are plastics engineered to have very small pores, just large enough to let some air through for breathing, but small enough to block viruses and other harmful particles. The plastic also carries an electric charge and is waterproof, which makes viruses and dust particles stick to the mask rather than going through and repels water altogether.

However, in many cases, it is not absolutely necessary to stop all the virus, but just the large clumps of it. While the virus that causes COVID19 is very small, often it does not just float around by itself. Instead, it clumps together in larger drops of liquid that can float in the air. These drops of liquid are pretty big and can be stopped by many more types of plastic (like those found in the more common, loose surgical masks), or even to some level by cotton fabrics if the fabric is thick and layered several times.

More "translucent" or "breathable fabrics" like silk or synthetic fabrics in athletic attire do not work however because they are specifically designed to have large holes for easier breathing and a "light" feeling. In all cases though, fibers found in clothing will not work as well as the special materials found in medical masks (including the loose surgical ones) - even if they have small pores, they do not have the added benefit of blocking the harmful particles with electric charge, and they will not be waterproof, meaning that water (and the virus dissolved in it) can seep through rather than being repelled chemically.

The virus is tiny, so tiny that you need something that is almost air-tight to stop virus particles. However, the virus quickly dies if it's not in water, which is why it needs to be in droplets, and the droplets are much larger. I am not sure if we know how effective the masks people are wearing are, but people who wear masks do seem to be at lower risk of contracting the virus. This may be because the mask-wearers are also practicing social distancing, but something that they are doing helps.

That's a hard question. In practical terms, the most important thing is to have a mask that you'll actually wear - it won't be under your chin because it's so hard to breathe through. Also, how many gaps are there around the edges where air is leaking in and out. Two fabric layers are needed.

I read about a study that said t-shirt fabric was great, and the person writing about the study said their t-shirts made bad masks. The reason was that good masks were made from heavy t-shirts, and bad masks were made from thin t-shirts.