Shabbos, Chapter 1, Mishnah 2. This mishnah is something of a departure from the laws of Shabbos. It discusses activities a person may not begin before davening Minchah - once the time for Minchah has arrived, and even half an hour before it arrives - out of concern that he will become caught up in what he is doing, the time for davening will pass, and he will not daven Minchah. The reason this mishnah appears here is that the next mishnah (Mishnah 3) deals with a similar idea: a person may not do certain things out of concern that he will forget and carry something on Shabbos. Both cases share one foundation: the concern that a person will become absorbed in what he is doing and forget.
Defining the time:
The mishnah opens: "Lo yeisheiv adam lifnei hasapar samuch l'Minchah" - a person should not sit down before a barber close to Minchah. 'Minchah' here means Minchah Gedolah - from six and a half hours into the day onward, which is the earliest time a person may daven Minchah. 'Close to Minchah' means half an hour before that, which in practice means from halachic midday onward.
The activities forbidden close to Minchah:
"Lo yeisheiv adam lifnei hasapar" - a person should not sit down for a haircut until he has davened. At first glance, what could go wrong in such a long stretch of time? The Gemara explains that the scissors might break and need to be repaired, and the whole process could become far more complicated than anticipated.
"Lo yikanes adam lamerchatz" - he should not enter the bathhouse before davening Minchah. Here too for the same reason: a person may get held up, and may even faint in the bathhouse from the heat and the like.
"V'lo l'bursaki" - he may not enter the tannery to process hides, lest a flaw be found in the tanning process and he have to begin the work all over again, something that could become complicated and take a great deal of time.
"V'lo le'echol" - he may not begin a substantial meal, since it is liable to go on for a long time.
"V'lo ladin" - a judge in beis din should not open a court case.
"V'im hischilu - ein mafsikin":
In all of these cases, if he transgressed and began - even though he was not permitted to do so - he is not required to interrupt what he is doing in order to daven, as long as there is still real time left for davening. If there is not enough time left, he must stop and daven. But after the fact, as long as time remains, he may continue his activity provided that he finishes in time to daven.
What counts as 'beginning' in each of these activities?
A haircut: from the moment the barber spreads the protective cloth around the person getting his hair cut.
The bathhouse: from the moment a person removes the last garment before entering, and according to another opinion, even from the removal of the first garment.
The tannery: from the moment he puts on the tanner's apron, the protective clothing he needs for the work.
A meal: from the moment he washes his hands.
A court case: from the moment the judges wrap themselves in their talleisim. And if they were already engaged in a case beforehand and were already wrapped in their talleisim, the beginning is counted from the moment the litigants start presenting their claims.
"Mafsikin l'kerias Shema v'ein mafsikin l'tefillah":
The mishnah closes with a statement that sounds similar to its opening, but in fact it is discussing an entirely different case: one interrupts in order to recite Shema, which is a Torah obligation, but one does not interrupt in order to daven tefillah, meaning Shemoneh Esrei. What is this referring to? The Gemara explains that it refers to a highly exceptional situation - people whose Torah is their occupation, whose entire involvement is Torah, such as Rabbi Shimon bar Yochai and his colleagues, who would learn Torah continuously without any interruption at all. In such a situation they were permitted to skip davening, provided that they fulfilled the Torah obligation of reciting Shema. Hence: "Mafsikin likros kerias Shema" - they would interrupt only for Shema, "v'ein mafsikin l'tefillah" - and they would not interrupt for Shemoneh Esrei. This, however, is an exceptional situation, one that in practice is not relevant to anyone in our generation.