Two men went up into the temple to pray: the one a Pharisee, and the other a publican. The Pharisee standing, prayed thus with himself: O God, I give thee thanks that I am not as the rest of men, extortioners, unjust, adulterers, as also is this publican. I fast twice in a week: I give tithes of all that I possess. And the publican, standing afar off, would not so much as lift up his eyes towards heaven; but struck his breast, saying: O god, be merciful to me a sinner. I say to you, this man went down into his house justified rather that the other: because every one that exalteth himself, shall be humbled: and he that humbleth himself, shall be exalted.
I remember when I first started attending the Latin Mass and observed someone striking their breast. I had never seen that before and was impressed by such a pious example.
According to Fisheaters the striking of the breast can occur at the following times:
Some Catholics strike their breast at the words "forgive us our trespasses" made formally during the Confiteor, the Nobis Quoque Peccatoribus, the Agnus Dei, and the Domine, Non Sum Dignus at the Mass, and informally any time to express regret.)
The striking of the breast is also done (three times) during the "Hail, Holy Queen" at the words, "Oh clement, oh loving, oh sweet Virgin Mary" and at the Angelus when the words, "And the Word was made Flesh, and Dwelt among us."
This custom not be peculiar to the Latin Mass but could be done (and should be) by any Catholic not because it makes us think better of ourselves but because it reminds us that we are sinners and should be pained at the thought of our sins.
Even if you do not have the opportunity to attend the Latin Mass do not be afraid claim the traditions that are rightfully yours. These are the traditions that will aid in your spiritual development and be an example to your children and other Catholics.