FAQ: How gas mask filters work, finally understood.
Since google searches for how the filters actually work in the game have come up with many varied responses, some of them contradictory, I thought I'd share the known facts here.

How much time is on a filter?

It varies, a random filter found in the game world will usually have about one minute of air time, whereas a filter bought from the merchant at market will have 10 minutes of air time. The time you have remaining on a filter is unaffected by your actions (running or jumping), but it is theorized that if Artyom is at "rest" (not breathing heavily, or even better standing near a fire to "calm" him) you will get more time on the filter when you swap one in. That is unconfirmed though, and doesn't seem to affect your total time available. It is important to note though that if your gas mask becomes cracked, your filter will get used up faster (unconfirmed).

How do I know how many filters I have remaining?

This information is not shown to you, and it seems to be irrelevant because you don't actually run out of filters, you run out of breathing time.

When should I change my filter?

Artyom's breathing will become labored when the equipped filter is about to expire. Artyom will automatically swap in a new filter when the equipped one expires, but as you cannot perform any actions while Artyom is doing this it's best to change your filter before entering combat.

How do I read Artyom's watch?

The red hand which moves counter-clockwise will be either in the green, yellow, or red areas of the watch. It is a countdown timer that shows how many minutes of air time you have remaining on the filter that you are wearing (e.g. if it's in the yellow you have less than 10 minutes, red less than 5, and when it reaches zero Artyom automatically changes his filter).

Help! My watch is broken, it always says I only have one minute left on my filter!

Some people say that their watch is "broken" and never moves out of the red, this is probably because you're out of "good" filters and have to keep using filters that only have one or two minutes of air. If you buy a filter you'll see that the watch works, but since you can only buy filters at Riga or Market you're probably going to be stuck with bum filters for the rest of the game.

What is the number next to the gas mask icon? I've been told it's how much air is left on the filter.

Not quite. Artyom's watch is the only measure of how much time is on a filter, what the number below the gas mask icon shows is how much TOTAL filter time you have among all your filters. If it says 10 you have 10 minutes of air, and it can be with 10 filters that have only one minute left, or 1 filter with 10 minutes.

Is there a maximum amount of filtered breathing time you can have?

Yes, no matter how many filters you buy you can't get the amount of time you have remaining over the ~40 minute range. The reason it is not a set number is because if you buy a filter that puts you over 40 minutes you then can't buy another one, e.g. if you were at 31 minutes and bought a filter it would put you at 41, but buying another wouldn't put you at a max it just would expend the money and you would not get a filter. I theorize that the max possible you can get is 49, but I haven't been able to test this.

Why can I pick up filters some times but not all the time?

This is not something that is entirely clear. It is possible that there is a maximum amount of filters you can carry, but there is no way to know. What it seems though is that if you have a certain amount of time left total on your filters, then it won't let you pick up new filters (I usually need to have under 25 minutes of time left before it will let me pick up filters found outside).

Why do the filters I find only last one minute, but the ones I buy last 10?

It's either for gameplay challenge reasons, or realism. At the time the game takes place all the filters should be expired. It's possible that the crappy filters you find are meant to be used or expired, whereas the ones you buy are recently made. Gas mask filters are somtimes made with charcoal, so it's not unrealistic to assume that it is within the Metro citizen's capabilities to create new gas mask filters. Also it's important to note that the filters bought at Market station last for 10 minutes, whereas the ones bought at Riga seem to last for less, somewhere around 4 minutes.

Where can I buy filters? None of the merchants sell them!

Only the merchants at Riga station and Market station sell filters. It's not known why, but it is certainly a frustration as you can't get new filters later in the game. Stock up early if you can. Note though that even the merchants who don't sell filters will still hawk, "Medkits! Filters! Knives!" since the sound file is not specific to the merchants.

Hope this helps some people.