Chapter 8, Mishnah 3, as well as the mishnayos that follow it, discuss different types of injured parties and how the five payments we learned about apply to them.
"Hachovel be'aviv uve'imo velo yatza meihem dam" - One who injures his father or his mother and no blood emerged from them:
The Mishnah is dealing specifically with an injury in which no blood emerged, because if blood did emerge - whether in a bleeding wound or a bruise beneath the surface of the skin - the one who caused the injury is liable to the death penalty. As long as no blood emerged, he is liable for all five payments. The same applies to one who injures his fellow on Yom Kippur: he too may be liable for all five payments.
The basis of this is the rule we have seen many times, according to which one who is liable to the death penalty is exempt from payments (and in the coming mishnayos we will return to discuss it from another angle). This rule does not apply to one who injures his parents when no blood emerged from them: since there is no possibility of the death penalty here, the one who caused the injury pays his parents the full payments.
"Vehachovel bachavero beYom Hakippurim" - And one who injures his fellow on Yom Kippur:
Even though drawing blood on Shabbos is a subcategory of a prohibited labor, and carries a potential death penalty, on Yom Kippur one who injures his fellow is liable to bring a chatas offering, and if he did it intentionally - he is liable to kares. However, there is no court-imposed death penalty here even with witnesses, and at most he would receive lashes. Since there is no execution, there is no exemption from the payments.
"Hachovel be'eved ivri" - One who injures a Jewish slave:
One who injures a Jewish slave is liable for all the payments. A Jewish slave is simply a regular Jew who sold his labor to his employer for six years. The only exception is the payment of sheves (unemployment compensation), when the one who caused the injury is the master - the employer or owner of that slave - because the economic output of the slave belongs to the master anyway, and there is no payment to be paid here.
"Hachovel be'eved kenaani shel acherim" - One who injures the Canaanite slave of others:
One who injures the Canaanite slave of his fellow, a non-Jewish slave, is liable for all the payments - and potentially all five payments - and the money ultimately goes to the owner of the slave, for the slave has no financial life of his own, and everything transfers to the master.
The dispute between Rabbi Yehudah and the Sages regarding payments for humiliation:
Rabbi Yehudah says: There is one exception, which is humiliation - one does not pay a slave payments for humiliation, meaning payment for degradation and emotional suffering. The reason is not as one might assume, but rather a technical reason derived from a scriptural decree: the verse from which the payments for humiliation are derived states "When men strive together... a man and his brother", and according to Rabbi Yehudah there is no relationship of brotherhood between a Jew and a non-Jewish slave, since there is no possibility of marriage between them - a Canaanite slave cannot enter the congregation of Israel as long as he is a slave. Since there is no concept of brotherhood, the verse does not allow for a humiliation payment.
And the Sages disagree, and the halachah follows them: A Canaanite slave is considered "your brother" - your brother regarding mitzvos, since he is obligated and commanded in mitzvos like a Jewish woman. Just as Jewish women and Jewish men are considered to have a relationship of brotherhood on a conceptual level, so too between a Jew and a Canaanite slave. Therefore, there are also payments for humiliation.
In summary: In this Mishnah, we learned three cases where the injurer is liable for all the payments: one who injures his parents when no blood is drawn, because there is no death penalty here that would exempt him from payment; one who injures his fellow on Yom Kippur, since his liability is a chatas or kares and at most malkus, and not a death penalty administered by the court; and one who injures a Hebrew slave - excluding the payment for sheves when the injurer is his master. We also learned the law of one who injures the Canaanite slave of his fellow, that the payments go to the owner, and the dispute between Rabbi Yehudah and the Chachamim regarding the payment for boshes, where the halachah follows the opinion of the Chachamim, who obligate him.
In the following Mishnayos, we will continue to discuss additional types of injurers and injured parties and the manner in which the laws of payments apply to them.