What is the Buddhist approach of admitting a mistake?

You must understand the karma and its result. It depends on the intention one make to do the karma and on what person one did this karma. It is not like in front of the jury plea guilty or not giving excuses. If you would like to apologize, directly do it to that person explaining and admitting sincerely with all your heart. If that person is no more, you can do it through an Arya (and/or Triple gems). Only thing is not to do bad karma, if done it apologize sincerely, even then it is not enough so that to achieve more good karma, work hard at insight meditation because good karma will make bad one less and less as the dew disappear as the sun shine. The stone that sink under water could be carried in the boat big enough to carry it.

Mistakes or misdeeds are so important that the Indian Buddhist monk and scholar Śāntideva (8th century CE) said in his śāstra (treatise) Śikṣāsamuccaya (Compendium of Teachings): "One law serves to summarize the whole of the Mahāyāna: The protection of all beings is accomplished through examination of one's own mistakes."

In the Tibetan traditions, the proper response to mistakes has been summarized in four steps called the four powers at least since the time of Tsongkhapa (14th century CE). The four powers are: (1) confessing or exposing one's misdeed before a witness or support, (2) feeling remorse for having committed the misdeed, (3) vowing not to repeat the misdeed in the future, and (4) taking action to remedy the effects of the misdeed. Exactly which actions should be taken to remedy the effects of the misdeed depends on the particular misdeed and the situation. There are many Tibetan commentaries that discuss these four steps.

Of course, the four powers are completely in accord with what Śākyamuni Buddha taught in the discourses. In the Sāmaññaphala Sutta (DN 2), the Buddha says to King Ajātasattu:

Yes, great king, a transgression overcame you in that you were so foolish, so muddle-headed, and so unskilled as to kill your father—a righteous man, a righteous king—for the sake of sovereign rulership. But because you see your transgression as such and make amends in accordance with the Dhamma, we accept your confession. For it is a cause of growth in the Dhamma and Discipline of the noble ones when, seeing a transgression as such, one makes amends in accordance with the Dhamma and exercises restraint in the future.

In the Ambalaṭṭhikārāhulovāda Sutta (MN 61), which another answer already cited, the Buddha says to Rāhula:

Having done a bodily action, you should reflect on it: "This bodily action I have done—did it lead to self-affliction, to the affliction of others, or to both? Was it an unskillful bodily action, with painful consequences, painful results?" If, on reflection, you know that it led to self-affliction, to the affliction of others, or to both; it was an unskillful bodily action with painful consequences, painful results, then you should confess it, reveal it, lay it open to the Teacher or to a knowledgeable companion in the holy life. Having confessed it, you should exercise restraint in the future.

And:

Rāhula, all those brahmans and contemplatives in the course of the past who purified their bodily actions, verbal actions, and mental actions, did it through repeated reflection on their bodily actions, verbal actions, and mental actions in just this way.

In a book titled Being Upright, the North American Zen teacher Reb Anderson quoted Śāntideva ("The protection of all beings is accomplished through examination of one's own mistakes") and commented:

Confession completely purifies our bodies and minds. Practicing it alone or together with others reawakens the heart of compassion and the appreciation of others' virtues. [...] To me, the word confession means "the admission of our actions." It is a literal and formal acknowledgment of our deliberate actions, or karma, in terms of body, speech, and thought. [...] Formal confession can be done alone, within your own heart, and that is certainly of great merit. However, in order to realize the complete and inconceivable function of formal confession, you must practice confession in the presence of another. When nonvirtue is brought forward out of the dark, you are more able to fully recognize and therefore confess it. Such confession before the face of another can be practiced in front of a statue of a Buddha or Bodhisattva, in front of a teacher, or in the presence of community members within a ritual context. There are an infinite number of possible ways of practicing formal confession. The essential point is that you honestly, sincerely, and thoroughly reveal your unwholesome actions.

The realization of the full, liberating function of formal confession must entail elements of regret and remorse. By "regret" I mean that you feel that you have made a mistake, wish that you had not committed the action, and sincerely intend to refrain from doing so again. Such regret and steadfast intention to refrain from nonvirtuous action is supported by feelings of remorse. Etymologically, remorse means "to bite again." The complete, purifying power of confession involves tasting again the bitterness of your nonvirtue. It may be useful to distinguish between remorse and disgust about your actions. Disgust is a feeling of "rejecting or loathing what you have done" and may undermine a steadfast sense of responsibility for your actions, whereas remorse has more the feeling of revisiting the scene of the crime and deepening your sense of responsibility. [...]

Bodhisattvas continuously confess their nonvirtue. Because delusion occurs moment after moment, confession must also be moment after moment. Bodhisattvas continuously see and admit their own delusions and nonvirtuous deeds. Less enlightened people confess less often. The most unenlightened and dangerous people are those who think that they never do any nonvirtuous deeds at all. The greatest darkness for the human mind is to believe that you never do anything wrong or hurtful or stupid. Conversely, continuous confession of nonvirtue opens the gate to great light. Confession of wrongdoing is an act of awakening.

In the excerpt from the book Being Upright that I just quoted above, Reb Anderson is talking about formal confession. But a little later in the book he notes that the Sōtō Zen tradition distinguishes between formal confession and formless confession:

The practice of formless or ultimate confession is based on formal confession but goes beyond it. In formal confession we fully recognize and acknowledge our karmic deeds. In formless confession, according to Dōgen, we "quietly explore the furthest reaches of the causes and conditions" of these actions. Formal confession refreshes and purifies us from the consequences of our self-centered actions of body, speech, and thought. Formless confession reaches to and removes the roots of these actions. Formal confession prepares us for meditation. Formless confession is the meditation process itself. It is being upright.

Formal confession is very important. And after formal confession there is the deeper process of formless confession, the process of exploring the furthest reaches of causes and conditions and consequences of all our activity. This process is also called vipassanā or vipaśyanā.