The photos do help a lot. Classic American Standard; we have something close in one bedroom and we love that seat with the quick-release hinges for easy cleaning!
Your flush valve has some appurtenances that suggest that it at one time had the old American Standard tilt-flush thingy on it. For that reason, your overflow riser (the vertical tube rising from the flush valve) doesn't have little "wings" down at the bottom on which to hook a flapper. So Fluidmaster gives you a little black piece that wraps around the overflow riser and slides down to the bottom and then you hook their flapper on to the little bars that protrude on the side. Which is what you (or the last person who installed replacement parts in it) already have done.
That Fluidmaster flapper has some fans and some detractors. For a generic toilet like this, you might find that a basic Korky flapper would work well. Any 2" Korky with a collar that goes around the overflow riser. Like this one, but in any color: red or black or this teal one at HD:
http://www.homedepot.com/p/Korky-Teal-2-in-Toilet-Tank-Flapper-63CM20/100164092 If you do go with the Korky, it talks about clipping the oval thing if you have a "plastic" flush valve. That's an arbitrary distinction; the real distinction is whether your flush valve has wings, which yours doesn't, so you would pull out all the black stuff around your flush valve -- the flapper and the adapter -- and slide the collar of the Korky over the existing overflow riser.
However, I would first try to make this Fluidmaster one work, and since I'm not there I can only observe a few things. First, you want the flapper to...well...flap down on the opening making a complete seal. That means that the black ring that goes around the overflow riser has to be set at a height that allows the flapper to lie flat on the valve opening. If it's too high it will leave a little opening close to the riser, too low it will leave a little opening away from the riser; this is exacerbated by the fact that the material in the Fluidmaster flapper seems to me to be a little less pliable than the Korky, so it has to be a little more precise. Just slide the collar thingy up or down a bit on the riser to make sure the flapper lies flat.
Second, Reach4 is right that that chain seems kinda tight. You want very little slack, but you want SOME slack, like a link or two. It looks like the clip is on the last link of the chain, but you seem to have the clip connected to the flush handle rod a little weirdly. If you can, clip it so the top of the clip (the short squarish part) is riding flat on top of the hole in the flush handle rod, and the rest of the clip is hanging down from there. When you do that, you will probably have too much slack, so you need to attach the clip to a lower link in the chain, which is easy to do. Requires a little dexterity, particularly if the chain and clip are wet, but it's doable.
Third, also because of the way the clip is attached to the flush handle rod, it looks like the flush handle rod or clip may be binding a bit against the rubber refill hose. Make sure it moves freely, and if you have to slide that clip around on the overflow riser, do so. You can also cut the refill hose with a scissors and give it a straighter route to the pastic clip on the top of the overflow riser, which you can position on the left side of the overflow riser (i.e. closer to the fill valve).
Hope this helps, and let us know how you make out. We want you to succeed, because this really isn't something you should have to spend $90 on a plumber call to fix.