Well:

F = Full fare First
A = Some sort of discounted First
P = Even more discounted First

Y = Full Fare Economy
all the rest = Economy in decreasing cost

Even though this particular search shows - F5 A5 P5 - this doesn't necessarily mean that there are at least 15 First Class seats open since a "higher" fare bucket will always include seats in a "lower" fare bucket - however, you can assume that there are at least 5 open First Class seats on this flight (and probably more since it seems like this airline shows only a maximum of 5 for First Class). Also, you can assume that much more than 9 seats are available in Economy since all the fare buckets show 9 and 9 is the maximum that the inventory will show for Economy seats.

Different airlines may show different codes for fare buckets, but for all F is First (either domestic or international), J or C for Business class (either international on a 2-class plane or middle class on a 3-class plane, Y for Economy.

None of these "free" tools show award seats or upgrade availability however.

So, for any search you'll come up with will show some sort of the following:

F _ _ JorC _ _ _ Y _ _ _ _ _ _ _ _ _ _ _ _