Kefula as is mentioned by the other answer, is a doubled over fold. You won't see these folds in store bought matzah as some of the comments have mentioned, because the ones that are baking look out for these folds and tend remove such matzot before sending the boxes out. Basically what happens is when one is placing the dough in the oven to be cooked, part of the dough ends up flopping over part of itself, like in the image below.
i understand that this picture has a fold on purpose, and that it isn't matzah, but this should give you a good visual representation for what is being discussed.
The real question is nefucha, which is commonly referred to as an air bubble, however i don't believe this is true. Nefucha is not prohibited by Yosef Caro in the Shulchan Arukh, but is prohibited by the Rema. But most people only quote the first part of what the Rema says, and conveniently leave out the words that immediately follow.
Shulchan Arukh 461:5
אם אפו חמץ עם מצה, לא נאסרה אלא אם כן נגעה בחמץ, ונוטל ממקום שנגעה כדי
נטילת מקום והשאר מתר
מצה נפוחה באמצעיתה אסורה, אבל אם עלה עליה קרום כדרך שעולה על הפת בשעת
אפיה, מתרת
If hametz was cooked with matzo, [the matzo] is not forbidden unless
it touched the hametz, and one must remove from the place of contact
to remove that spot, but the rest is permitted. Rem"a:...Matzo that is
swollen in the middle is forbidden. If the upper skin rose like it
does with bread during baking, it is permitted.
So clearly the Rema is speaking about a swelling in the middle that is forbidden, but he clearly states this swelling is NOT air bubbles, or any type of rising that is normal to the bread process. Unfortunately, despite the Rema's clear words, in the past 200 years most Ashkenazim rule that a nefucha is an air bubble. But if the Rema isn't prohibiting air bubbles, then what is he prohibiting? There was an incident regarding this issue that happened a few hundred years ago between a traveling rabbi from Jerusalem to a Yemenite community.
Yemenite Rabbi Mansura’s comment on this issue was written into the margin of a Shulhan Arukh with the Pri Hadash commentary. Here we cite some of what he said:
I would not have had to write this, were it not for a certain incident
that happened. Namely, in the year 132 (for documents, i.e. 5579 or
1819) Rabbi David Nahmias, an emissary from the Holy City, appeared in
our holy community and was put up in the home of the notable dayyan
and holy man of G-d, my rabbi and teacher Rabbi Joseph Karah, z”l.
When they were setting the table, Rabbi David Nahmias made fun of us
and cast doubt on our kashruth, saying that (Heaven forfend) we were
eating hametz on Passover, since there were bubbles on the surface of
the loaf, and this he said was the puffed up matzah of which Maharil
wrote. Hence he did not want to eat.
In other words, Rabbi David Nahmias of Jerusalem refused to eat the Yemenite Jews’ matzah because he thought the matzah which they made fell into the category of matzah which is puffed in the center, and such matzah is forbidden. Rabbi Nahmias erred in not understanding that Rema’s reference to “ matzah which is puffed in the middle” pertained to thick matzah, since when it is thick it might not be baked through thoroughly. With Yemenite matzah, however, the law of matzah which is puffed in the middle does not apply since the Yemenite Jews bake extremely thin matzahs, and the bubbly surface on the matzahs which they served were not an indication of it being “matzah which is puffed in the middle.” Regarding this Rabbi Mansura wrote:
In my humble opinion it seems to me that these words apply to their
matzahs, which are one tefah thick … but according to our custom,
there is no reason for apprehension, since if you take ten of the
matzahs that we bake and pile them one on the other, the stack will
still not be one tefah thick.
Rabbi Mansura criticized the way matzah was baked in the Land of Israel, in contrast to the way it was made in Yemen:
You should see our beautiful cool water, and such fast kneading, that
even in your land it is not equaled. It is also baked speedily, in
one minute. Not so in your land, for we have heard complaints that
you bake an extremely thick loaf, without hearing the flames of the
furnace, and that it remains stuck to the walls of the oven for a long
time; hence one suspects that it is way too far away from the fire for
it to be baked in the blink of an eye, as our bread is made.
Moreover, to refute the words of the aforementioned Rabbi David (meaning Rabbi David Nahmias), who boasted of the practice in his land and criticized the matzah-baking custom of Sanaa, it is written in the work of Rabbi Pri Hadash, par. 459, 2, that the practice in the holy city of Jerusalem is not proper, for their dough ferments, and there is no one who properly understands this (see the above reference to the Pri Hadash).
Source: http://www.biu.ac.il/JH/Parasha/eng/pesah/gim.html
So according to the Rabbis of Yemen, a nefucha is a swelling in the dough that is caused by attempting to cook too thick of a matzah in an oven that's not hot enough to cook the matzah all the way through. Which makes sense since the Rema speaks of a swelling in the middle, which is the last part of the matzah to be cooked as the heat radiates from outside to in, and if your oven isn't hot enough you could end up with a center that isn't cooking and therefore starts to rise.
But despite clearly distinguishing these two ideas, most modern Poskim rule that one has to poke holes in their matzah to prevent air bubbles, which they call nefucha. But either way, in modern matzah that is made with perforations and is cooked incredibly thin and dry, one will never run across a true nefucha, and the perforations usually prevent any air bubbles, so one typically has nothing to worry about when buying packaged matzah that is a hard cracker.
